Short answer
Yes. Several passages in the Bible fit the Qur'an closely. In Deuteronomy, God (﷾) promises a prophet like Moses and says He will put His words in his mouth. The Qur'an is exactly that: God (﷾) speaks in it as "I", and the Prophet Muhammad (ﷺ) repeated it word for word. Jesus spoke of "the Spirit of truth" to come, who would not speak of himself but only what he heard. The Qur'an says the same of the Prophet (ﷺ): he does not speak from his own desire, only what is revealed to him. This is why Muslims hold that the Bible itself points to the Qur'an.
Full explanation
"I will put my words in his mouth"
Deuteronomy is the fifth book of the Torah, the first five books of the Bible. In it God (﷾) promised Moses a prophet like him "from among their brethren". Brethren means brothers, the kin of Israel, and they include the sons of Ishmael. This prophet would speak God's (﷾) words in God's (﷾) name. The Torah itself says no prophet like Moses arose again in Israel. The Qur'an speaks in just this way. God (﷾) speaks in the first person ("Verily, I am Allah", "I am your Lord"), with no "God said" in front. More than three hundred times God (﷾) tells him "Say" (qul), giving the Prophet (ﷺ) the words to speak. He never claimed to be God, and Islam forbids the claim. So these are God's (﷾) words in a man's mouth.
“I will raise them up a Prophet from among their brethren, like unto thee, and will put my words in his mouth; and he shall speak unto them all that I shall command him. And it shall come to pass, that whosoever will not hearken unto my words which he shall speak in my name, I will require it of him.”
“And command thou the people, saying, Ye are to pass through the coast of your brethren the children of Esau, which dwell in Seir; and they shall be afraid of you: take ye good heed unto yourselves therefore:”
“And there arose not a prophet since in Israel like unto Moses, whom the Lord knew face to face,”
“Verily, I am Allah: There is no god but I: So serve thou Me (only), and establish regular prayer for celebrating My praise.”
“Verily, this brotherhood of yours is a single brotherhood, and I am your Lord and Cherisher: therefore serve Me (and no other).”
“Say: He is Allah, the One and Only; Allah, the Eternal, Absolute; He begetteth not, nor is He begotten; And there is none like unto Him.”
The Book speaks to him
The Qur'an gives the Prophet (ﷺ) orders, such as to rise and pray by night. It even corrects him, for frowning at a blind man and for forbidding himself what God (﷾) had made lawful. He once moved his tongue quickly to hold on to each new verse. The Qur'an told him not to hurry, because gathering and reciting it was God's (﷾) work. A man does not address himself like this. It is the voice of the one who put the words in his mouth.
“O thou folded in garments! Stand (to prayer) by night, but not all night,- Half of it,- or a little less, Or a little more; and recite the Qur'an in slow, measured rhythmic tones. Soon shall We send down to thee a weighty Message.”
“(The Prophet) frowned and turned away, Because there came to him the blind man (interrupting). But what could tell thee but that perchance he might grow (in spiritual understanding)?- Or that he might receive admonition, and the teaching might profit him? As to one who regards Himself as self-sufficient, To him dost thou attend; Though it is no blame to thee if he grow not (in spiritual understanding). But as to him who came to thee striving earnestly, And with fear (in his heart), Of him wast thou unmindful.”
“O Prophet! Why holdest thou to be forbidden that which Allah has made lawful to thee? Thou seekest to please thy consorts. But Allah is Oft-Forgiving, Most Merciful.”
“Move not thy tongue concerning the (Qur'an) to make haste therewith. It is for Us to collect it and to promulgate it: But when We have promulgated it, follow thou its recital (as promulgated): Nay more, it is for Us to explain it (and make it clear):”
Ibn Abbas explains that the Prophet (ﷺ) used to find the revelation heavy and would move his lips quickly with it, until Allah revealed that he should not hurry his tongue: gathering and reciting the Qur'an was Allah's task. After that he listened while Gabriel recited, and then recited it as Gabriel had.
God's (﷾) words, or men's words about God (﷾)?
Christians hold that the Bible was written by many men under God's (﷾) inspiration, each in his own words and style. Christian scholars say so openly. W. Graham Scroggie, a Scottish Baptist minister, wrote in 1922 that its books passed through the minds of men and bear their style. Kenneth Cragg, an Anglican scholar of Islam, wrote in 1956 that the Gospels came through the mind of the church. Luke, who wrote one of the New Testament Gospels, says he researched it from those before him. Paul's letters end with his personal greetings to friends. The Qur'an has no such verses. The Prophet's (ﷺ) own words are kept apart, in the hadith (the reports of what he said and did). Think of two teachers. One tells a student, "teach them what I taught you". The student passes on the ideas in his own words. The other hands his student a book and says, "read it word for word". The student passes on the teacher's own words. The Bible's books passed through the minds of men, as in the first case. The Qur'an, where God (﷾) speaks as "I", is of the second.
Scroggie, a Scottish Baptist minister, wrote that the Bible is human as well as divine: its books passed through the minds of men, were written in the language of men and by their hands, and bear the marks of their style.
Cragg, an Anglican scholar of Islam and later an assistant bishop in Jerusalem, contrasted the New Testament with the Qur'an: the Gospels, he wrote, involve condensation and editing, and have come through the mind of the church behind their authors.
“Forasmuch as many have taken in hand to set forth in order a declaration of those things which are most surely believed among us, Even as they delivered them unto us, which from the beginning were eyewitnesses, and ministers of the word; It seemed good to me also, having had perfect understanding of all things from the very first, to write unto thee in order, most excellent Theophilus,”
“All that are with me salute thee. Greet them that love us in the faith. Grace be with you all. Amen.”
“Salute Prisca and Aquila, and the household of Onesiphorus.”
“Greet all the brethren with an holy kiss.”
“Verily, I am Allah: There is no god but I: So serve thou Me (only), and establish regular prayer for celebrating My praise.”
"He shall not speak of himself"
Jesus told his disciples that the Spirit of truth would come after him. This one would not speak of himself, but only what he heard. The Qur'an says the same of the Prophet (ﷺ). He does not speak from his own desire, and what he brings is revelation sent down to him. The one Jesus describes hears and then speaks. He guides into all truth, and he tells of things to come. These are the marks of a prophet who receives revelation. They describe the Prophet (ﷺ), who passed on the Qur'an as he heard it.
“Howbeit when he, the Spirit of truth, is come, he will guide you into all truth: for he shall not speak of himself; but whatsoever he shall hear, that shall he speak: and he will shew you things to come.”
“Nor does he say (aught) of (his own) Desire. It is no less than inspiration sent down to him:”
A new song from the villages of Kedar
Isaiah calls for "a new song" to the Lord from the ends of the earth. He names the wilderness and "the villages that Kedar doth inhabit". Kedar was the second son of Ishmael, and his name was carried by a people of northern Arabia. Among the places Isaiah names for this praise is Kedar in Arabia, where the Qur'an was revealed.
“Sing unto the Lord a new song, and his praise from the end of the earth, ye that go down to the sea, and all that is therein; the isles, and the inhabitants thereof. Let the wilderness and the cities thereof lift up their voice, the villages that Kedar doth inhabit: let the inhabitants of the rock sing, let them shout from the top of the mountains.”
“And these are the names of the sons of Ishmael, by their names, according to their generations: the firstborn of Ishmael, Nebajoth; and Kedar, and Adbeel, and Mibsam,”
The answer to Abraham's prayer
The Qur'an tells how Abraham and Ishmael prayed while raising the Ka'ba (the House of God (﷾) in Mecca). They asked for a messenger from among their own descendants, who would recite God's (﷾) signs to them and teach them the scripture. The Prophet (ﷺ), a descendant of Ishmael, is that messenger. The Qur'an is the scripture he recited.
“And remember Abraham and Isma'il raised the foundations of the House (With this prayer): "Our Lord! Accept (this service) from us: For Thou art the All-Hearing, the All-knowing.”
“"Our Lord! make of us Muslims, bowing to Thy (Will), and of our progeny a people Muslim, bowing to Thy (will); and show us our place for the celebration of (due) rites; and turn unto us (in Mercy); for Thou art the Oft-Returning, Most Merciful.”
“"Our Lord! send amongst them a Messenger of their own, who shall rehearse Thy Signs to them and instruct them in scripture and wisdom, and sanctify them: For Thou art the Exalted in Might, the Wise."”
Verse by verse
Deuteronomy 18:18-19 — Torah, Devarim, chapter 18, verses 18-19
“I will raise them up a Prophet from among their brethren, like unto thee, and will put my words in his mouth; and he shall speak unto them all that I shall command him. And it shall come to pass, that whosoever will not hearken unto my words which he shall speak in my name, I will require it of him.”
Cited for
Peter says in Acts that this prophet like Moses is Jesus. The promise is fulfilled in him, not in anyone after him.
The response
The promise is of a prophet "from among their brethren", Israel's kin. The Torah itself says no prophet like Moses arose again in Israel. In John's Gospel the priests ask John the Baptist whether he is the Christ, and then separately whether he is "that prophet". So they held the two to be different people. And the sign Deuteronomy gives is God's (﷾) own words, put in the prophet's mouth and spoken in God's (﷾) name. That is the form of the whole Qur'an.
Where it lands — Deuteronomy's prophet comes from Israel's brethren and brings God's (﷾) own words; John's Gospel keeps him apart from the Christ, and the Qur'an fits the promise.
Deuteronomy 18:15 — Torah, Devarim, chapter 18, verse 15
“The Lord thy God will raise up unto thee a Prophet from the midst of thee, of thy brethren, like unto me; unto him ye shall hearken;”
Cited for
"Brethren" in Deuteronomy means fellow Israelites (17:15; 18:2), and verse 15 says "from the midst of thee".
The response
The word covers both: Deuteronomy 2:4 calls the sons of Esau "your brethren". In verse 18, where God (﷾) Himself speaks, the words are only "from among their brethren". And the Torah closes with no such prophet having arisen in Israel.
Where it lands — The word allows Israel's kin, and God's (﷾) own wording in verse 18 leaves the door open to them.
John 14:26 — Gospel of John, chapter 14, verse 26
“But the Comforter, which is the Holy Ghost, whom the Father will send in my name, he shall teach you all things, and bring all things to your remembrance, whatsoever I have said unto you.”
Cited for
Jesus himself names the Comforter: "which is the Holy Ghost". So the Paraclete (the Greek word for the Comforter) is the Holy Spirit.
The response
An early Syriac translation of this verse has only "the Spirit". And the Gospels do not fit the identification. The Paraclete could not come unless Jesus went away to the one who sent him. Yet the Holy Spirit filled John the Baptist from the womb and came down on Jesus at the Jordan. While Jesus was still teaching, he said the Father gives the Holy Spirit to those who ask. And in John he gives the Holy Ghost to his disciples before he has gone up to the Father. Acts says he was taken up forty days later.
Where it lands — The Holy Spirit was already present and given before Jesus went to the Father; the Paraclete could come only after he had gone.
Perspectives
Christian
Christians answer that the prophet like Moses in Deuteronomy is Jesus, as Peter, a disciple of Jesus, says in Acts. They read the Spirit of truth in John as the Holy Spirit.
In Acts, Peter applies the promise of a prophet like Moses to Jesus.
Rebuttal
Peter applies it to Jesus, but in John the priests ask separately about the Christ and "that prophet", as two different people. Jesus said his words were "not of myself" (John 14:10), but the New Testament Gospels give them in their writers' words. The Qur'an is God's (﷾) own speech, word for word, as Deuteronomy describes.
Christians say "brethren" in Deuteronomy means fellow Israelites, and that verse 15 says "from the midst of thee".
Rebuttal
The word covers both: Deuteronomy 2:4 calls the sons of Esau "your brethren". In verse 18, where God (﷾) Himself speaks, the words are only "from among their brethren". The Torah then closes with no such prophet having arisen in Israel.
Christians read the Spirit of truth as the Holy Spirit, whom John 14:26 names.
Rebuttal
John 14:26 does say "the Holy Ghost", but an early Syriac translation has only "the Spirit". Before Jesus left, the Holy Spirit had filled John the Baptist and come down on Jesus. The Paraclete could come only after he went. Jesus promised "another Comforter". And 1 John uses the same word, parakletos, for Jesus himself: the one to come is like him.
The Islamic perspective
The Bible promises a prophet who would speak God's (﷾) own words. This is fulfilled in the Qur'an, revealed to the Prophet Muhammad (ﷺ).
Deuteronomy promises a prophet like Moses. God (﷾) will put His words in his mouth, and he will speak in God's (﷾) name.
The Qur'an speaks as God (﷾) in the first person. More than three hundred times it commands the Prophet (ﷺ) to "Say" what it gives him. This is God's (﷾) words put in a prophet's mouth.
The Qur'an commands the Prophet (ﷺ) and corrects him. It corrects him for frowning at a blind man, and for forbidding himself what God (﷾) had made lawful.
A book that commands and rebukes the man who recites it speaks from outside him.
Christian scholars describe the Bible as written in the words of men, and it contains personal greetings of Paul. The Qur'an contains no words of the Prophet's (ﷺ) own.
Jesus described one to come who would not speak of himself but only what he heard. The Qur'an says the Prophet (ﷺ) does not speak from his own desire but from revelation.
Isaiah links a new song of praise with the villages of Kedar, Ishmael's son.
The Qur'an presents the Prophet (ﷺ) as the messenger Abraham and Ishmael prayed for.
Taken together, these passages foretell the revelation of the Qur'an.