Short answer
Yes. Jesus called himself a prophet. On his way to Jerusalem he said, "it cannot be that a prophet perish out of Jerusalem". When his home town rejected him, he said, "A prophet is not without honour, save in his own country". The crowds called him "Jesus the prophet of Nazareth". In Luke's last chapter, two of his followers still call him "a prophet mighty in deed and word". He never corrected anyone for calling him a prophet. He said his teaching was not his own, but came from the one who sent him. Peter, his leading disciple, preached him as "a man approved of God". Christians say he was more than a prophet, and those replies are answered below. Islam agrees with Jesus' own words. In the Qur'an he says that God (﷾) "made me a prophet", and the Qur'an calls him a messenger, like the messengers before him.
Full explanation
Jesus counted himself among the prophets
Some Pharisees, members of a strict Jewish religious party, warned Jesus that Herod, the ruler of Galilee, wanted to kill him. He answered that he would keep on his road, because a prophet does not perish outside Jerusalem. In the next breath he mourns the city that kills the prophets and stones those sent to it. He counted himself among the prophets that Jerusalem turned against. The Qur'an says his enemies did not in fact kill him. Earlier, when the people of his home town Nazareth rejected him, he answered with a saying about prophets. The Gospels of Matthew, Mark and Luke all record it, and John's Gospel says Jesus himself "testified" it.
“Nevertheless I must walk to day, and to morrow, and the day following: for it cannot be that a prophet perish out of Jerusalem.”
“O Jerusalem, Jerusalem, which killest the prophets, and stonest them that are sent unto thee; how often would I have gathered thy children together, as a hen doth gather her brood under her wings, and ye would not!”
“That they said (in boast), "We killed Christ Jesus the son of Mary, the Messenger of Allah";- but they killed him not, nor crucified him, but so it was made to appear to them, and those who differ therein are full of doubts, with no (certain) knowledge, but only conjecture to follow, for of a surety they killed him not:-”
“And they were offended in him. But Jesus said unto them, A prophet is not without honour, save in his own country, and in his own house.”
“But Jesus said unto them, A prophet is not without honour, but in his own country, and among his own kin, and in his own house.”
“And he said, Verily I say unto you, No prophet is accepted in his own country.”
“For Jesus himself testified, that a prophet hath no honour in his own country.”
Others called him a prophet, and he let it stand
Many others called him a prophet: the crowds in Jerusalem, the crowd at the village of Nain, and a woman from Samaria. So did the blind man he healed. In Luke's last chapter two followers describe him to a stranger as a prophet, and Luke says the stranger was Jesus. He rebukes them for being slow to believe the prophets, not for calling him one. Nowhere in the Gospels does he correct anyone for the title.
“And the multitude said, This is Jesus the prophet of Nazareth of Galilee.”
“And there came a fear on all: and they glorified God, saying, That a great prophet is risen up among us; and, That God hath visited his people.”
“The woman saith unto him, Sir, I perceive that thou art a prophet.”
“They say unto the blind man again, What sayest thou of him, that he hath opened thine eyes? He said, He is a prophet.”
“And he said unto them, What things? And they said unto him, Concerning Jesus of Nazareth, which was a prophet mighty in deed and word before God and all the people:”
“Then he said unto them, O fools, and slow of heart to believe all that the prophets have spoken:”
Peter preached a man approved of God (﷾)
The book of Acts, in the New Testament, tells the story of the first Christians. In it Peter, the leading disciple of Jesus, preached Jesus as "a man approved of God". Through him, Peter said, God (﷾) did the miracles. In the same sermon he said that God (﷾) "hath made" Jesus "both Lord and Christ". ("Christ" means "anointed": one chosen by God (﷾) for a task.) So even his highest titles were given to him by God (﷾). In the Temple at Jerusalem, Peter called Jesus God's servant, whom God (﷾) had glorified. The one whom God (﷾) approves, makes Lord and glorifies is God's (﷾) servant, not God.
“Ye men of Israel, hear these words; Jesus of Nazareth, a man approved of God among you by miracles and wonders and signs, which God did by him in the midst of you, as ye yourselves also know:”
“Therefore let all the house of Israel know assuredly, that God hath made that same Jesus, whom ye have crucified, both Lord and Christ.”
“The God of Abraham, Isaac, and Jacob, the God of our fathers, has glorified his Servant Jesus, whom you delivered up, and denied in the presence of Pilate, when he had determined to release him.”
He spoke as one sent
Jesus spoke of his message the way every messenger does. His teaching, he said, was not his own but belonged to the one who sent him. The words his hearers heard were the Father's. In prayer he called the Father "the only true God", and himself the one God (﷾) had sent.
“Jesus answered them, and said, My doctrine is not mine, but his that sent me.”
“He that loveth me not keepeth not my sayings: and the word which ye hear is not mine, but the Father’s which sent me.”
“And this is life eternal, that they might know thee the only true God, and Jesus Christ, whom thou hast sent.”
What Christians reply
Christians agree that Jesus was a prophet, but say he was more. They call him prophet, priest and king. The Letter to the Hebrews, by an unnamed early Christian writer, says God (﷾) spoke through the prophets and at last through his Son. Jesus said something greater than the prophet Jonah and King Solomon was present, and "Before Abraham was, I am". Many also say he is the prophet like Moses that Moses himself foretold. Each reply is answered verse by verse below.
“God, who at sundry times and in divers manners spake in time past unto the fathers by the prophets, Hath in these last days spoken unto us by his Son, whom he hath appointed heir of all things, by whom also he made the worlds;”
“The men of Nineveh shall rise in judgment with this generation, and shall condemn it: because they repented at the preaching of Jonas; and, behold, a greater than Jonas is here. The queen of the south shall rise up in the judgment with this generation, and shall condemn it: for she came from the uttermost parts of the earth to hear the wisdom of Solomon; and, behold, a greater than Solomon is here.”
“Jesus said unto them, Verily, verily, I say unto you, Before Abraham was, I am.”
“For Moses truly said unto the fathers, A prophet shall the Lord your God raise up unto you of your brethren, like unto me; him shall ye hear in all things whatsoever he shall say unto you.”
Following Jesus means believing what he said about himself
Jesus called himself a prophet more than once. He said, "A prophet is not without honour, save in his own country". He also said, "it cannot be that a prophet perish out of Jerusalem". He spoke of himself as one sent. His teaching, he said, was "his that sent me", and in prayer he spoke of "Jesus Christ, whom thou hast sent". He never said "I am God" or "worship me". He told Satan, "Thou shalt worship the Lord thy God, and him only shalt thou serve". Most churches call him a prophet too. But the creed of Nicaea is a statement of belief written by church bishops in 325. It puts at its centre a title he never claimed: "true God of true God". Christians honour Jesus and say they follow him. Muslims hold that following him means believing what he said about himself. In the Qur'an he says it himself: "I am indeed a servant of Allah: He hath given me revelation and made me a prophet".
“And they were offended in him. But Jesus said unto them, A prophet is not without honour, save in his own country, and in his own house.”
“Nevertheless I must walk to day, and to morrow, and the day following: for it cannot be that a prophet perish out of Jerusalem.”
“Jesus answered them, and said, My doctrine is not mine, but his that sent me.”
“And this is life eternal, that they might know thee the only true God, and Jesus Christ, whom thou hast sent.”
“Then saith Jesus unto him, Get thee hence, Satan: for it is written, Thou shalt worship the Lord thy God, and him only shalt thou serve.”
The creed opens: "We believe in one God, the Father Almighty, maker of all things visible and invisible". The council confessed the Son as begotten of the Father, true God of true God, and "of one substance" (homoousios) with the Father, and condemned those who said there was a time when he was not.
“He said: "I am indeed a servant of Allah: He hath given me revelation and made me a prophet;”
The Qur'an says the same
The Qur'an agrees with what the Gospels show. Speaking as a baby in the cradle, Jesus says Allah (﷾) made him a prophet. The Qur'an calls him a messenger, like the messengers who passed away before him. It also has him tell the Children of Israel that he is the messenger of Allah (﷾) to them. He brings them good news of a messenger to come after him, named Ahmad, which Muslims know as a name of Muhammad (ﷺ).
“He said: "I am indeed a servant of Allah: He hath given me revelation and made me a prophet;”
“Christ the son of Mary was no more than a messenger; many were the messengers that passed away before him. His mother was a woman of truth. They had both to eat their (daily) food. See how Allah doth make His signs clear to them; yet see in what ways they are deluded away from the truth!”
“And remember, Jesus, the son of Mary, said: "O Children of Israel! I am the messenger of Allah (sent) to you, confirming the Law (which came) before me, and giving Glad Tidings of a Messenger to come after me, whose name shall be Ahmad." But when he came to them with Clear Signs, they said, "this is evident sorcery!"”
Verse by verse
Hebrews 1:1-2 — Epistle to the Hebrews, chapter 1, verses 1-2
“God, who at sundry times and in divers manners spake in time past unto the fathers by the prophets, Hath in these last days spoken unto us by his Son, whom he hath appointed heir of all things, by whom also he made the worlds;”
Cited for
God spoke through the prophets, and last of all through his Son. So the Son stands above every prophet.
The response
Hebrews is a letter by an unnamed writer, not a saying of Jesus. In the Bible "son" is a title of God's (﷾) favour. God (﷾) calls Israel "my son, even my firstborn", and Luke calls Adam "the son of God". Paul, a Christian missionary of the first century, says all who are led by God's (﷾) Spirit are "the sons of God". The same letter calls Jesus "the Apostle" (one sent), faithful to the one who appointed him, "as also Moses was". Verse 8 calls the Son "O God", but it quotes Psalm 45, a wedding song for a king of Israel. The very next verse says to the Son, "God, even thy God, hath anointed thee with the oil of gladness above thy fellows".
Where it lands — Hebrews itself ranks Jesus beside Moses, as one appointed by God (﷾). The title "Son" does not lift him out of the ranks of the sent.
Matthew 16:16 — Gospel of Matthew, chapter 16, verse 16
“And Simon Peter answered and said, Thou art the Christ, the Son of the living God.”
Cited for
People said Jesus was "one of the prophets". Peter said more: "the Christ, the Son of the living God". Jesus blessed him for it.
The response
"Christ" means anointed, that is, marked out by God (﷾) for a task. Luke records Peter's answer as "The Christ of God". Peter later preached how "God anointed Jesus of Nazareth", "for God was with him". The Bible even calls Cyrus, a Persian king, the Lord's "anointed".
Where it lands — Peter confessed a role that God (﷾) gave to Jesus. The one anointed and the God (﷾) who anoints him are two.
Matthew 12:41-42 — Gospel of Matthew, chapter 12, verses 41-42
“The men of Nineveh shall rise in judgment with this generation, and shall condemn it: because they repented at the preaching of Jonas; and, behold, a greater than Jonas is here. The queen of the south shall rise up in the judgment with this generation, and shall condemn it: for she came from the uttermost parts of the earth to hear the wisdom of Solomon; and, behold, a greater than Solomon is here.”
Cited for
Jesus said that something "greater than Jonas", the prophet Jonah, was here. He said something "greater than Solomon", the wise king, was here too. No mere prophet could say that.
The response
Jesus said John the Baptist was "more than a prophet". He also said that the least in the kingdom of God (﷾) is greater than John. Nobody takes that to make John, or the least believer, God (﷾).
Where it lands — Being greater among God's (﷾) messengers keeps a man among them. Greater among the sent is still sent.
Revelation 17:14 — Revelation of John, chapter 17, verse 14
“These shall make war with the Lamb, and the Lamb shall overcome them: for he is Lord of lords, and King of kings: and they that are with him are called, and chosen, and faithful.”
Cited for
Jesus is prophet, priest and king. The Letter to the Hebrews calls him High Priest and the book of Revelation calls him "King of kings". No mere prophet holds all three offices.
The response
The Bible gives each title to men. Melchizedek, who met Abraham, was king of Salem and "priest of the most high God". The Persian king Artaxerxes is called "king of kings". Nebuchadnezzar, king of Babylon, is also "a king of kings", because God (﷾) gave him his kingdom. Revelation says believers are made "kings and priests". Hebrews calls Jesus "the Apostle", a word that means one sent, and says God (﷾) appointed him high priest. In Revelation Jesus speaks from heaven and says "my God" four times in one verse.
Where it lands — Prophet, priest and king: each of these titles is shared with men. The titles add honour, not divinity.
John 8:58 — Gospel of John, chapter 8, verse 58
“Jesus said unto them, Verily, verily, I say unto you, Before Abraham was, I am.”
Cited for
Jesus said "Before Abraham was, I am". No mere prophet existed before Abraham.
The response
"I am" is an everyday phrase. In John 9:9 the healed blind man says the same Greek words, ego eimi ("I am"), of himself. In John chapter 8 Jesus says "Before Abraham was, I am". In that same chapter he also says, "I do nothing of myself; but as my Father hath taught me". He calls himself "a man that hath told you the truth, which I have heard of God". God (﷾) told Jeremiah, "Before I formed thee in the belly I knew thee", and ordained him a prophet.
Where it lands — A prophet can exist in God's (﷾) plan before his birth. The speaker of John 8 calls himself a man taught by God (﷾).
John 1:14 — Gospel of John, chapter 1, verse 14
“And the Word was made flesh, and dwelt among us, (and we beheld his glory, the glory as of the only begotten of the Father,) full of grace and truth.”
Cited for
Christians say Jesus was a true prophet in his human nature and God the Son in his divine nature. John writes that "the Word", whom Christians take to be God the Son, "was made flesh", that is, became a man.
The response
John 1:14 is the writer's own comment, not a saying of Jesus. "Two natures" is the language of the Council of Chalcedon, a meeting of church bishops in 451, not of any New Testament Gospel. Chalcedon says Christ is one person, "one and the same Son". That one Son says "my Father is greater than I". He calls himself "a man that hath told you the truth, which I have heard of God". Just before he is taken up to heaven, he still says the times are in the Father's power.
Where it lands — The one person who speaks in the Gospels puts God (﷾) above himself, from first to last. A prophet who says this of himself is not God.
Acts 3:22 — Acts of the Apostles, chapter 3, verse 22
“For Moses truly said unto the fathers, A prophet shall the Lord your God raise up unto you of your brethren, like unto me; him shall ye hear in all things whatsoever he shall say unto you.”
Cited for
Peter says the prophet like Moses is Jesus, so Deuteronomy 18 is already fulfilled and points to no later prophet.
The response
The book of Acts records how Peter read the promise. But the Torah, the five books of Moses, gives its own test: the prophet must be like Moses. Jesus is unlike Moses on point after point. In John's Gospel, the Jews of Jesus' day treated "that prophet" and "the Christ" as two different men. Christians and Muslims agree that Jesus is the Christ.
Where it lands — The prophet like Moses and the Christ were expected as two men. Jesus is the Christ, and the prophet like Moses is Muhammad (ﷺ), as the page on Deuteronomy 18 sets out.
John 9:38 — Gospel of John, chapter 9, verse 38
“And he said, Lord, I believe. And he worshipped him.”
Cited for
The blind man who called Jesus a prophet later believed on "the Son of God" and worshipped him.
The response
The Greek word here is proskuneo, which means to bow down. It is also used of bowing to a master. In a story Jesus told, a servant "fell down, and worshipped him". The Bible calls Israel God's son, Adam "the son of God", and all led by God's (﷾) Spirit "the sons of God".
Where it lands — Bowing and the title "Son of God" are both given to men in the Bible. Neither makes the one who receives them God.
Perspectives
Mainstream Christian
Jesus was a true prophet, but more than a prophet. He is also priest and king, and he is God the Son made man.
Jesus holds three offices: prophet, priest and king. The Letter to the Hebrews calls him High Priest, and the book of Revelation calls him "King of kings".
Rebuttal
The Bible gives each of these titles to men. Melchizedek, who met Abraham, was both a king and a priest. Nebuchadnezzar, king of Babylon, was "a king of kings" because God (﷾) gave him his kingdom. Believers are made "kings and priests". Hebrews says God (﷾) appointed Jesus high priest. In Revelation the one called "King of kings" speaks of "my God" four times in one verse.
The Letter to the Hebrews says God spoke through the prophets, and in these last days through his Son. So the Son stands above every prophet.
Rebuttal
Verse 8 of the same chapter calls the Son "O God". But it is quoting Psalm 45, a wedding song for a king of Israel. The chapter also says God (﷾) "appointed" the Son heir. It says God (﷾), as "thy God", anointed him "above thy fellows". One who has a God is not God. The same letter calls Jesus "the Apostle" (one sent), faithful to the one who appointed him, "as also Moses was".
People said Jesus was one of the prophets. Peter called him "the Christ, the Son of the living God", and Jesus blessed him.
Rebuttal
"Christ" means anointed, that is, marked out by God (﷾) for a task. Luke gives Peter's words as "The Christ of God", and Peter later preached that God (﷾) anointed Jesus. The Bible calls even the Persian king Cyrus the Lord's anointed. "Son of God" is a title the Bible also gives to Israel, to Adam and to all led by God's (﷾) Spirit. The Qur'an too calls Jesus the Christ, and in the same breath "a messenger of Allah".
Jesus said "a greater than Jonas is here", meaning the prophet Jonah, and "a greater than Solomon is here". So he is more than a prophet.
Rebuttal
Jesus called John the Baptist "more than a prophet", and said the least in the kingdom of God (﷾) is greater than John. Being greater than a prophet does not make John, or the least believer, God (﷾). Greater among the sent is still sent.
A prophet can also be God. Christ is one person in two natures, one human and one divine. So he was a prophet as a man and God as the eternal Son.
Rebuttal
No New Testament Gospel says "two natures". The Council of Chalcedon, a meeting of church bishops, said it in 451. The council calls Christ "one and the same Son". That one Son says "my Father is greater than I". In Mark 13:32 it is "the Son", set against "the Father", who does not know the hour. He calls himself "a man" who heard truth from God (﷾). In Acts, just before he is taken up, he still leaves the timing of events to the Father, as a prophet does.
Jesus is the prophet like Moses whom Deuteronomy 18 foretold, as Peter preached in Acts 3:22. So, they say, the promise has no later prophet in view.
Rebuttal
The book of Acts records how Peter read the promise. But the Torah, the five books of Moses, gives its own test: the prophet must be like Moses. Jesus is unlike Moses on point after point. In John's Gospel, the Jews of Jesus' day treated "that prophet" and "the Christ" as two different men. John the Baptist denied being either, and the crowds divided over which one Jesus was. Christians and Muslims agree that Jesus is the Christ. So the prophet like Moses, raised "from among their brethren", is another man, as the page on Deuteronomy 18 sets out.
Jesus said "Before Abraham was, I am", claiming to exist before Abraham, which no mere prophet can do.
Rebuttal
"I am" is an everyday phrase. In John 9:9 the healed blind man says the same Greek words, ego eimi ("I am"), of himself. In John chapter 8, his "I am he" comes with "I do nothing of myself", and he calls himself "a man" taught by God (﷾). God (﷾) knew Jeremiah before forming him in the belly and ordained him a prophet. So a prophet can exist in God's (﷾) plan before birth.
In John, those who first call Jesus a prophet go on to see more. The man born blind later believes on "the Son of God" and worships him.
Rebuttal
The Greek word for "worshipped" here is proskuneo, which means to bow down. It is also the bowing given to a master. In a story Jesus told, a servant "fell down, and worshipped him". "Son of God" is a title the Bible also gives to Israel, to Adam and to all who are led by God's (﷾) Spirit.
Christians say that Jesus is a prophet and also God. So, they say, the creeds do not set aside what he said about himself.
Rebuttal
The creed of Nicaea is the bishops' wording, written in 325, not his. In the Gospels he calls himself a prophet and one sent. He never says "I am God" or "worship me". He tells Satan, "Thou shalt worship the Lord thy God, and him only shalt thou serve". Muslims hold that to follow him is to take him at his word.
The Islamic perspective
Jesus was a prophet and messenger of God (﷾), as he said himself. The Gospels and the Qur'an agree on it.
Jesus called himself a prophet, in Luke 13:33 and in the saying about a prophet in his own country.
The crowds and his own followers called him a prophet, and the Gospels never show him correcting them.
Peter preached Jesus as "a man approved of God", whom God (﷾) made "both Lord and Christ". He also preached him as God's (﷾) servant.
Jesus said his teaching and his words came from the Father who sent him.
Jesus called himself a prophet and one sent. The Gospels never have him say "I am God" or "worship me".
The Qur'an teaches that Jesus was a prophet and messenger of Allah (﷾), like the messengers before him.