""Your father Abraham rejoiced as he looked forward to my coming. He saw it and was glad." The people said, "You aren't even fifty years old. How can you say you have seen Abraham?" Jesus answered, "I tell you the truth, before Abraham was even born, I Am!""
.that you may know that the Son of Man has authority on earth to forgive sins"--he then said to the paralytic--"Rise, pick up your bed and go home." And he rose and went home. When the crowds saw it, they were afraid, and they glorified God..."7 He said to them, "You are from below; I am from above. You are of this world; I am not of this world. I told you that you would die in your sins, for unless you believe that I am he you will die in your sins."8
"When John heard in prison what Christ was doing, he sent his disciples to ask him, 'Are you the one who was to come, or should we expect someone else?' Jesus replied, 'Go back and report to John what you hear and see; The blind receive sight, the lame walk, those who have leprosy are cured, the deaf hear, the dead are raised, and the good news is preached to the poor.'"
The jews saw His claims as being he was God
Furthermore, Jesus made an unmistakable claim to deity before the chief priests and the whole Sanhedrin. Caiaphas the high priest asked Him: “‘Are you the Christ, the Son of the Blessed One?’ ‘I am,’ said Jesus. ‘And you will see the Son of Man sitting at the right hand of the Mighty One and coming on the clouds of heaven’” (Mark14:61–62). A biblically illiterate person might well have missed the import of Jesus’ words. Caiaphas and the council, however, did not. They knew that in saying he was “the Son of Man” who would come “on the clouds of heaven” he was making an overt reference to the Son of Man in Daniel’s prophecy (Dan. 7:13–14). In doing so, He was not only claiming to be the preexistent Sovereign of the universe but prophesying that He would vindicate His claim by judging the very court that was now condemning Him. Moreover, by combining Daniel’s prophecy with David’s proclamation in Psalm 110, Jesus was claiming that He would sit upon the throne of Israel’s God and share God’s very glory. To students of the Old Testament this was the height of “blasphemy,” thus “they all condemned him as worthy of death” (Mark14:64–65).
http://www.equip.org/article/did-jesus-claim-to-be-god/
Yes, Jesus did do that. Make unmistakable claim to being deity (God) but saying so in many ways included what else you also pointed towards;
invoking there being the Great I Am whom when Moses asked --