I was once asked this question by a Jehovah Witness to prove if the Jews understood that Jesus was claiming to be God. I affirm that they did while the Jehovah Witnesses deny it.
I'm going to approach this by saying the following. Whether or not the Jews are correctly or incorrectly understanding Jesus is not the issue. The issue is what was it that Jesus "SAID" that caused the Jews to say He was claiming to be God?
John 5:16, "And for this reason the Jews were persecuting Jesus, because He was doing thins on the Sabbath." What Jesus did was heal on man on the Sabbath. Jesus says at John 5:17, "My Father is working until now, and I Myself am working."
John 5:18, "For this cause therefore the Jews were seeking all the more to KILL Him, because (or why?) He not only was breaking the Sabbath, but also was calling God His own Father, making Himself equal with God."
Jesus claimed to be the Son of God in the sense that no other believer is of God's essence. Look what Jesus says at Luke 2:49 when He was a boy. "Why is it that you were looking for Me? Did you not know that I had to be in MY FATHER'S HOUSE?"
What did Jesus say at John 8:58 that upset the Jews? "Truly, truly, I say to you, before Abraham was born, I am." Jesus was saying to the Jews that before Abraham's existence He himself was existing. This is also confirmed at John 1:15 by John the Baptist. "John bore witness of Him, and cried out, saying, "This was He of whom I said, He who comes after me has a higher rank than I, FOR HE EXISTED BEFORE ME."
So how did the Jews reply to what Jesus stated, "Therefore they picked up stones to throw at Him; but Jesus his Himself, and went out of the temple." Did you notice Jesus made His statement right there in the Temple.
How about what Jesus says at John 10:30, He literally says in the Greek, "I and the Father WE are one." Jesus distinguishes the "I" from the "Father" and uses the plural verb "are" denoting "we are." Asserting their unity of essence or nature as identical. Hence John 10:31, "The Jews took up stone AGAIN to stone Him."
The Jews understood Jesus correctly. There could be no mistake about His meaning. "Again" refers to John 8:59 at which time the Jews also attempted to stone Him for blasphemy.
At John 19::7, "The Jews answered him, "We have a law, and by that law He ought to die because He made Himself out the Son of God." Matthew's trial record gets into more detail than the trial record here at John 19.
The high priest Caiaphas at Matthew 26:63 ask Jesus a question? "I adjure You (swear to us) by the living God, that You tell us whether You are the Christ, the Son of God?" Caiaphas was clearly asking if Jesus was ONE person; "The Christ, the Son of (God)." Obviously Caiaphas did not believe that Jesus was "The Christ the Son of (God)"; it was a rhetorical question intended to trap Jesus into committing blasphemy.
Jesus, calling Himself the Son of God is not blasphemy because He really WAS the Son of God. This is confirmed or proved by multiple Scriptures. Notice also how the the claim "Son of God," "equal to God," "making Yourself out God" are used interchangeably by the Jews. The high priest Caiaphas finally says at Matthew 26:65, "He has blasphemed! What further need do we have of witnesses? Behold, you have now heard the blasphemy."
In conclusion, I find that there is an abundance of evidence that the Jews knew exactly that Jesus was clearly claiming to be God. Moreover, there is clear evidence throughout the New Testament that others stated He was God who were Jews as well.
As a side note! I could have proved the Jews knew Jesus was claiming to be God by writing the following 16 words or so and calling it a day. "If the Jews misunderstood Jesus, why did they bring up the Law at John 19:7?" The law that Jesus "supposedly" broke can be found at Leviticus 24:16. Bottom line, the Jews did not believe Jesus Christ.
The following is from "Jews For Jesus." https://jewsforjesus.org/publications/issues/issues-v14-n02/the-claims-of-jesus/
This is another site that backs up what Jesus was claiming. https://www.oneplace.com/ministries/bible-answer-man/read/articles/did-jesus-claim-to-be-god-by-hank-hanegraaff-8825.html
Jesus claiming to be God would have been blasphemy.
Jesus did not make such a claim.
Jesus said he was the Son of God and it was the Jewish religious leaders who in their desperate search for a reason to kill him made the leap that claiming sonship was the same as the much more serious blasphemous claiming to be God.
Jesus attempted to point out that he was not guilty of blasphemy in John 10:32-38
Jesus said to them, ‘I have done many good works for you to see, works from my Father; for which of these are you stoning me?’ The Jews answered him, ‘We are not stoning you for doing a good work but for blasphemy: you are only a man and you claim to be God’. Jesus answered: ‘Is it not written in your Law: I said, you are gods? So the Law uses the word gods of those to whom the word of God was addressed, and scripture cannot be rejected. Yet you say to someone the Father has consecrated and sent into the world, “You are blaspheming”, because he says, “I am the Son of God”. If I am not doing my Father’s work, there is no need to believe me; but if I am doing it, then even if you refuse to believe in me, at least believe in the work I do; then you will know for sure that the Father is in me and I am in the Father’”
Basically Jesus said the scriptures called certain men gods so what’s the problem with me saying I am God’s son?
No the Jews were not right in accusing Jesus of blasphemy. He had to remain sinless according to the law given to the Jews by Jehovah through Moses. Claiming to be God would have broken the very law Jesus came to perfectly fulfill.
Jesus was born under the Law of Moses.
But when the time had fully come, God sent his Son, born of a woman, born under law, to redeem those under law, that we might receive the full rights of sons (Galatians 4:4,5).
Jesus was also sinless under the law
Can any of you prove me guilty of sin? If I am telling the truth, why don't you believe me (John 8:46)
Jesus, as the promised Messiah, came and fulfilled the Law of Moses. He testified.
Do not think that I came to destroy the Law or the Prophets. I did not come to destroy but to fulfill (Matthew 5:17).
Jesus is the only person who has ever kept the law perfectly. He did everything the law required, never once breaking any of its commandments. Because He was sinless, Jesus was able to meet the requirements of the law to be the perfect sacrifice. His death redeemed humanity from the curse of the law. (Source)
Whether one believes Jesus is the created Son of Jehovah as Jehovah’s Witnesses do or believe that he is part of a triune Godhead,while he was on earth as a Jew he did not commit a sin according to the law covenant.
The charge of blasphemy was a false charge brought by the religious leadership in an effort to make his execution seem justified to the common Jew.
And of course remember that the Romans cared not about the charge of blasphemy anyway. That is why a second lie had to be trumped up against him that would trigger the Romans to execute him. That charge was sedition. The Jews succeeded in getting Jesus executed by saying He had claimed to be a king. This too was false accusation because Jesus had spoken of his heavenly kingdom and not of overthrowing the Roman government.