Jesus said "I AM."

John 4:25 The woman said, “I know that Messiah” (called Christ) “is coming. When he comes, he will explain everything to us.”

26 Then Jesus declared, “I, the one speaking to you—I am he.”

HE refers to Messiah as asked in the question.  The Messiah was a reference to the promised anointed One who was believed to be God.

Jewish theology refered to the passage in Daniel 7:13 to define the Messiah:

“In my vision at night I looked, and there before me was one like a son of man,[a] coming with the clouds of heaven. He approached the Ancient of Days and was led into his presence. 14 He was given authority, glory and sovereign power; all nations and peoples of every language worshiped him. His dominion is an everlasting dominion that will not pass away, and his kingdom is one that will never be destroyed."

Notice it says, "All nations and peoples of every language worshiped him."

This is an obvious claim of deity.

In John 8:58, Jesus said:
"“Very truly I tell you,” Jesus answered, “before Abraham was born, I am!”

The Jews took this as a claim to deity as you can see in the next verse.
"At this, they picked up stones to stone him, but Jesus hid himself, slipping away from the temple grounds."

It wasn't just because he claimes pre-existence.  Jesus already claimed pre-existence in verse 56, where he says:
"Your father Abraham rejoiced at the thought of seeing my day; he saw it and was glad.”

The reason they thought Jesus was claiming divinity was because he said, "I am."
In Hebrew, this would have been YWYH, God's name, which means, "I am."

The mighty God of the universe calls Himself I AM. In Exodus 3:14, the Bible says, “And God said unto Moses, I AM THAT I AM” (KJV). Moses had said, “When I go down to Egypt and the people ask me who sent me, what shall I tell them” (Cf. Exodus 3:13). God said, “You tell them that I AM sent you.”

Eight more times in the Gospel of John He uses the expression, “I am.”
Jesus said, “I am the bread of life. He who comes to Me shall never hunger” (John 6:35).
Jesus said, “I am the living bread which came down from heaven” (Cf. John 6:33). “If anyone eats of this bread, he will live forever” (John 6:51)
Jesus said, “I am the light of the world” (John 8:12)
Jesus said, “I am the door” (John 10:9)
Jesus said, “I am the true vine” (John 15:1)
Jesus said, “I am the good shepherd. The good shepherd gives His life for the sheep” (John 10:11)
Jesus said in John 11:25, “I am the resurrection and the life.”
Jesus said, “I am the way, the truth, and the life. No one comes to the Father except through Me” (John 14:6)

In all these "I am" statements, Jesus shows he is the way to God. Will you come to God through Jesus Christ?