The short answer to this question is: yes, he did. However, we believe in a triune God: God the Father, God the Son, and God the Holy Spirit. So Jesus, who is God the Son, could not and did not say ‘I am God’ in the sense that God is not outside Him. Jesus, God the Son, was on earth, but God the Father still was in heaven.
We see a clear biblical proof for this in Matthew 3:16-17: “And when Jesus was baptized, immediately he went up from the water, and behold, the heavens were opened to him, and he saw the Spirit of God descending like a dove and coming to rest on him; and behold, a voice from heaven said, “This is my beloved Son, with whom I am well pleased.” In these verses we see God the Son, Jesus, on the earth; God the Father speaking from heaven; and God the Spirit descending from heaven on the Son. Because of this rich relationship within the Trinity, it is not surprising that Jesus never uttered the words ‘I am God’.
However, Jesus clearly taught that he was God the Son. Though some people nowadays doubt it, during his time on earth people did not. The Jews want to arrest Jesus for “you, being a man, make yourself God” (John 10:33). Jesus did not contradict them. In the end, it is exactly this claim of Jesus that leads to his crucifixion. The high priest asks him: “tell us if you are the Christ, the Son of God” (Matthew 26:63). Jesus confirms it. The high priest calls it blasphemy, because Jesus makes himself the equal of God.
It is not strange that the Jews and the high priest understood that Jesus claimed to be God. He had given reason enough for that in his teaching.
In John 8:58 Jesus said: “Truly, truly, I say to you, before Abraham was, I am.” He not just says that he existed before Abraham existed 2000 years ago. He also calls himself ‘I am’, which is the name of God.
In John 10:30 Jesus said: “I and the Father are one.”
Jesus used several other saying with ‘I am’ in which he claimed to be equal to God. An example of this is “I am the light of the world. Whoever follows me will not walk in darkness, but will have the light of life” (John 8:12). This can never be said about just a man. Only God is the light of the world. Only God can give the light of life.
Jesus was sentenced to death and nailed to the cross for claiming to be God. But because he actually was God, death could not hold him. On the third day he rose from the grave, proving his claims to be God in the flesh.
After his resurrection, just before he went to heaven, Jesus commanded his disciples to baptise people ‘in the name of the Father and of the Son and of the Holy Spirit” (Matthew 28:19). Again, we see the Trinity here in the words of Jesus. Father, Son, and Holy Spirit are equal and are equally God.
Yes, Jesus claimed to be God. Jesus is God. That is great news, because only God himself could rightfully make the promise that Jesus made “I am with you always, to the end of the age” (Matthew 28:20). Jesus is eternal. Jesus is omnipresent. That is why he can take care of us and bless us even today.