Short answer
No, not on the Gospels' own account of his last night. He said his soul was sorrowful even unto death. He fell on his face and prayed that the cup, the suffering ahead, might pass from him. Luke says his sweat fell like drops of blood, and an angel came to strengthen him. Earlier that night he had told his disciples to buy swords. Matthew and Mark say that on the cross he cried out, asking why God (﷾) had forsaken him. John does have Jesus say that he lays down his life of himself. But even a willing substitute would not make it just to punish the innocent for the guilty. Islam's answer is that God (﷾) saved him: the Qur'an says his enemies did not kill him or crucify him.
Full explanation
What Christians believe
Christians hold that Jesus came to die for the sins of the world and gave his life freely. They point to John, where Jesus says no man takes his life from him. He says he lays it down of himself. In John he also tells Peter to put away his sword. He says he must drink the cup the Father has given him.
“No man taketh it from me, but I lay it down of myself. I have power to lay it down, and I have power to take it again. This commandment have I received of my Father.”
“Then said Jesus unto Peter, Put up thy sword into the sheath: the cup which my Father hath given me, shall I not drink it?”
He prepared for danger
At the last supper Jesus told any disciple without a sword to sell his garment and buy one. They showed him two swords, and he said it was enough. That is the order of a man expecting danger. In a place called Gethsemane, as Matthew tells it, he told Peter and the two sons of Zebedee, "tarry ye here, and watch with me".
“Then said he unto them, But now, he that hath a purse, let him take it, and likewise his scrip: and he that hath no sword, let him sell his garment, and buy one. For I say unto you, that this that is written must yet be accomplished in me, And he was reckoned among the transgressors: for the things concerning me have an end. And they said, Lord, behold, here are two swords. And he said unto them, It is enough.”
“Then saith he unto them, My soul is exceeding sorrowful, even unto death: tarry ye here, and watch with me.”
He prayed to be spared
The first three Gospels, Matthew, Mark and Luke, all record the same prayer. Jesus asked the Father to take the cup away from him, if it were possible. His own will was for the cup to pass. He then yielded to God's (﷾) will. Only Luke says that he was in agony and that his sweat fell like great drops of blood. Luke also says an angel came from heaven to strengthen him. Some of the oldest manuscripts of Luke lack these two verses. Even John, which has no prayer in Gethsemane, has him say, "Now is my soul troubled" as the hour came.
“And he went a little further, and fell on his face, and prayed, saying, O my Father, if it be possible, let this cup pass from me: nevertheless not as I will, but as thou wilt.”
“And he said, Abba, Father, all things are possible unto thee; take away this cup from me: nevertheless not what I will, but what thou wilt.”
“Saying, Father, if thou be willing, remove this cup from me: nevertheless not my will, but thine, be done.”
“And there appeared an angel unto him from heaven, strengthening him. And being in an agony he prayed more earnestly: and his sweat was as it were great drops of blood falling down to the ground.”
“Now is my soul troubled; and what shall I say? Father, save me from this hour: but for this cause came I unto this hour.”
He was heard
Hebrews says that Jesus prayed with strong crying and tears to the One who was able to save him from death. On the cross, as Matthew and Mark tell it, his cry is a question: why has God (﷾) forsaken him? A man who prays with tears to be saved from death is asking to be spared. Hebrews says he was heard. Christians point out that the same letter says God (﷾) brought Jesus again from the dead. But a prayer to be saved from death, when heard, is answered by rescue.
“Who in the days of his flesh, when he had offered up prayers and supplications with strong crying and tears unto him that was able to save him from death, and was heard in that he feared;”
“And about the ninth hour Jesus cried with a loud voice, saying, Eli, Eli, lama sabachthani? that is to say, My God, my God, why hast thou forsaken me?”
“Now the God of peace, that brought again from the dead our Lord Jesus, that great shepherd of the sheep, through the blood of the everlasting covenant,”
A plot, not a sacrifice
The Gospels do not describe a ritual offering. The chief priests met to take Jesus by trickery and kill him. Pilate, the Roman governor, said he found no fault in him. By the Gospels' own account, what they describe is the murder of an innocent man.
“Then assembled together the chief priests, and the scribes, and the elders of the people, unto the palace of the high priest, who was called Caiaphas, And consulted that they might take Jesus by subtilty, and kill him.”
“Then said Pilate to the chief priests and to the people, I find no fault in this man.”
A willing substitute is still unjust
The Law of Moses says every man shall be put to death for his own sin. Ezekiel, a prophet of the sixth century BC, says the son shall not bear the father's sin. Moses once offered to be blotted out of God's (﷾) book in Israel's place. God (﷾) refused: the one who sinned is the one He blots out. A Christian may say God (﷾) refused Moses only because Moses was a sinner too. But God (﷾) said nothing about Moses. He gave a rule: whoever has sinned is the one He blots out. Ezekiel makes repentance the way back, with no one punished in the sinner's place. The Qur'an states that no bearer of burdens can bear the burden of another. Willingness does not change this.
“The fathers shall not be put to death for the children, neither shall the children be put to death for the fathers: every man shall be put to death for his own sin.”
“The soul that sinneth, it shall die. The son shall not bear the iniquity of the father, neither shall the father bear the iniquity of the son: the righteousness of the righteous shall be upon him, and the wickedness of the wicked shall be upon him.”
“Yet now, if thou wilt forgive their sin—; and if not, blot me, I pray thee, out of thy book which thou hast written. And the Lord said unto Moses, Whosoever hath sinned against me, him will I blot out of my book.”
“But if the wicked will turn from all his sins that he hath committed, and keep all my statutes, and do that which is lawful and right, he shall surely live, he shall not die. All his transgressions that he hath committed, they shall not be mentioned unto him: in his righteousness that he hath done he shall live.”
“Say: "Shall I seek for (my) Cherisher other than Allah, when He is the Cherisher of all things (that exist)? Every soul draws the meed of its acts on none but itself: no bearer of burdens can bear the burden of another. Your goal in the end is towards Allah: He will tell you the truth of the things wherein ye disputed."”
What the Qur'an says happened
The Qur'an states that his enemies did not kill him and did not crucify him. It was made to appear so to them, and Allah (﷾) raised him up to Himself. Muslims measure the Gospels by the Qur'an, which confirms the earlier scripture and stands as judge over it. A prayer to be spared fits what the Qur'an says, and the prayer was answered.
“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:-”
“Nay, Allah raised him up unto Himself; and Allah is Exalted in Power, Wise;-”
“To thee We sent the Scripture in truth, confirming the scripture that came before it, and guarding it in safety: so judge between them by what Allah hath revealed, and follow not their vain desires, diverging from the Truth that hath come to thee. To each among you have we prescribed a law and an open way. If Allah had so willed, He would have made you a single people, but (His plan is) to test you in what He hath given you: so strive as in a race in all virtues. The goal of you all is to Allah; it is He that will show you the truth of the matters in which ye dispute;”
Verse by verse
John 10:18 — Gospel of John, chapter 10, verse 18
“No man taketh it from me, but I lay it down of myself. I have power to lay it down, and I have power to take it again. This commandment have I received of my Father.”
Cited for
Jesus says no man takes his life from him and he lays it down of himself, so his death was his own free choice.
The response
Matthew, Mark and Luke show the same man on his face in Gethsemane, asking for the cup to be taken away. Hebrews says he prayed to be saved from death and was heard. And even a willing substitute cannot make it just to punish the innocent for the guilty. God (﷾) refused Moses' offer to be blotted out for Israel.
Where it lands — One saying in John does not undo three Gospels' account of his prayer, and willingness would not make the exchange just.
Hebrews 5:7 — Epistle to the Hebrews, chapter 5, verse 7
“Who in the days of his flesh, when he had offered up prayers and supplications with strong crying and tears unto him that was able to save him from death, and was heard in that he feared;”
Cited for
"Save him from death" means "out of death". God heard the prayer by raising Jesus after he died.
The response
The prayer in Hebrews, with strong crying and tears, matches Gethsemane, where he asked that the cup pass from him. He did not ask to die and be raised. The writer of Hebrews also believed that Jesus died and was brought again from the dead, as the churches of his day taught. But the report he keeps of the prayer is plain. Jesus asked the One able to save him from death, and he was heard. On its own words, that is a prayer answered by rescue, which is what the Qur'an says happened.
Where it lands — Hebrews says the prayer to be saved from death was heard, and a prayer to be spared death is answered only if death is spared.
Luke 22:42 — Gospel of Luke, chapter 22, verse 42
“Saying, Father, if thou be willing, remove this cup from me: nevertheless not my will, but thine, be done.”
Cited for
"Not my will, but thine, be done" shows Jesus freely submitting to his death.
The response
The prayer shows that his own will was for the cup to pass from him. That is the first thing he asks. In Matthew he goes back to pray three times, still speaking of the cup. Then he submits his will to God's (﷾), as a servant does. A man who prays on his face to be spared is not going gladly.
Where it lands — The prayer shows a servant asking to be spared and yielding to God (﷾), not a man who wished to die.
John 12:27 — Gospel of John, chapter 12, verse 27
“Now is my soul troubled; and what shall I say? Father, save me from this hour: but for this cause came I unto this hour.”
Cited for
"For this cause came I unto this hour" shows that Jesus came into the world in order to die.
The response
In the King James text shown here, Jesus says in the same breath, "Father, save me from this hour". Other Bibles print it as a question that he turns down. Either way, his soul is "troubled". In the first three Gospels he prays, on his face, for the cup to pass. Hebrews says his prayer to the One able to save him from death was heard. The Qur'an states that God (﷾) raised him up and his enemies did not kill him.
Where it lands — Even in John his soul is troubled at the hour, and in the other three Gospels he asks to be spared.
Luke 22:36-38 — Gospel of Luke, chapter 22, verses 36-38
“Then said he unto them, But now, he that hath a purse, let him take it, and likewise his scrip: and he that hath no sword, let him sell his garment, and buy one. For I say unto you, that this that is written must yet be accomplished in me, And he was reckoned among the transgressors: for the things concerning me have an end. And they said, Lord, behold, here are two swords. And he said unto them, It is enough.”
Cited for
Jesus asked for swords only so that the scripture "he was reckoned among the transgressors" would be fulfilled. When Peter used one, Jesus stopped him.
The response
Verse 37 gives a reason for the swords. It does not explain his prayer, on his face and three times over, that the cup pass from him. The prayer is the heart of it. By the time the sword was drawn, Jesus had already yielded his will to God's (﷾) in prayer.
Where it lands — Whatever the swords were for, his prayer that night shows a man asking to be spared, not one seeking death.
John 18:11 — Gospel of John, chapter 18, verse 11
“Then said Jesus unto Peter, Put up thy sword into the sheath: the cup which my Father hath given me, shall I not drink it?”
Cited for
Jesus tells Peter to put away his sword because he must drink the cup the Father has given him, so he chose the cross.
The response
That same night, Matthew, Mark and Luke say, he had prayed for that same cup to pass from him. Matthew says he prayed on his face, three times. When the sword was drawn, Matthew has Jesus say he could ask for twelve legions of angels. He also asks how else the scriptures would be fulfilled. These are the words of a servant yielding to God's (﷾) will after asking to be spared, not of a man who wished to die. Hebrews says his prayer to the One able to save him from death was heard. The Qur'an states that Allah (﷾) raised him up to Himself.
Where it lands — Submitting to God (﷾) after asking to be spared is not the same as wanting to die.
Perspectives
Mainstream Christian
Christians say Jesus came into the world to give his life as a sacrifice for sin. They say he went to the cross freely, in obedience to the Father.
In John, Jesus says that no man takes his life from him, and that he lays it down of himself.
Rebuttal
John does record these words, but Matthew, Mark and Luke show him praying for the cup to pass. Hebrews says his prayer to be saved from death was heard. Even a willing substitute cannot make punishing the innocent just, as God's (﷾) refusal of Moses' offer shows.
In John, Jesus tells Peter to sheathe his sword. He asks whether he should not drink the cup the Father has given him.
Rebuttal
John does record this. But that same night Matthew, Mark and Luke say he prayed for the same cup to pass, and Matthew has him on his face. Submitting to God (﷾) after asking to be spared is not the same as wanting to die.
The Islamic perspective
Jesus did not seek death. The Gospels show him praying to be saved from it, and the Qur'an says God (﷾) saved him. His enemies did not kill him or crucify him. And no one can carry another's sin, willing or not.
Matthew, Mark and Luke all record Jesus praying for the cup to be taken from him.
Jesus said his soul was sorrowful even unto death. He fell on his face to pray that the cup might pass from him.
Hebrews says he prayed with tears to the One able to save him from death, and was heard.
His order to buy swords fits a man expecting danger, not one walking to a planned death.
The Law and the prophets forbid punishing one person for another's sin. God (﷾) refused Moses when he offered himself in Israel's place.
The Qur'an teaches that no soul bears another's burden. It also teaches that Jesus was neither killed nor crucified but raised by Allah (﷾) to Himself.