Does Jewish history record Jesus's saying "I am God" or similar utterance?

The closest thing we have to a Jewish text in which Jesus makes claims about himself - whether as the messiah, the son of God, or anything else - is Josephus' Antiquities of the Jews.

Josephus was a well educated, relatively well-to-do Jewish man, who became an officer in the Jewish rebel army during the Jewish Revolt. He led a unit of soldiers until they were captured by the Romans; Josephus wisely chose to become a turncoat in order to save his neck. He began to work with the Romans as an advisor, ultimately going so far as to attempt to negotiate an end to the prolonged siege of Jerusalem. The negotiations failed, the city was sacked, the Temple was torn down, and everyone inside the city walls was slaughtered. Many Jews have held this treachery against Josephus ever since.

After the end of the war, the Jews were scattered and diminished in numbers, and Josephus, probably realizing that he was persona non grata, moved to Rome, where he became a court historian to the Emperor. He was contracted to write a history of the revolt, and later, a general history of the Jewish people. These records are incredibly important, and provide an insight into a period of time which is difficult to study due to the mass carnage and destruction caused by the revolt.

Josephus' works contain two alleged references to Jesus, but one of them is almost universally regarded as a forgery inserted by later Christian scribes who were tasked with copying the manuscript. I won't include the spurious passage here, but I can summarize the reasons for it being dismissed as a fake. Everything we know about Josephus suggests that he remained a committed Jew his entire life, and had no relationship with the followers of Jesus. Yet the spurious passage would have us believe that Josephus believed that Jesus was indeed the messiah, he performed miracles, he was divine, and he was resurrected. There is absolutely no chance whatsoever that Josephus would have written such a thing. If he had actually believed these things, he would have joined the Jesus Movement, rather than remaining a Jew. The scholarly consensus is nearly unanimous in saying that this passage, known as the Testimonium Flavianum, is a later forgery.

The other reference to Jesus in Josephus, on the other hand, is widely regarded as authentic. For starters, it doesn't say anything that would be problematic for a Jewish author; it also rings true because it mentions Jesus only in passing, and he is not the focus of the story.

A little background information will be useful to provide the necessary context of the story. It appears that the only person who was authorized to order an execution in Roman Judaea was the proconsul - in Jesus' time, this office was held by Pontius Pilate. The Romans understood that the political situation in Judaea was complex, and frequently quite tense. Therefore, they chose to use the Temple priests as intermediaries between the Roman government and the Jewish people in the province. The priests were appointed by the Romans, usually through shady exchanges of money and influence.

Josephus' account describes a chain of events set in motion by the death of the proconsul, a man named Porcius Festus. It took some time for word of the proconsul's death to reach Rome - probably a couple of months - and a similar amount of time for the replacement, Lucceius Albinus, to arrive in Jerusalem.

In the interim, the High Priest, whose name was Ananus, decided to exploit the power vacuum by disposing of some troublemakers. It seems that Ananus had come into conflict with James, the brother of Jesus, who had replaced Jesus as the leader of the disciples following the crucifixion. It is thought that Ananus may have tried to convince the Romans to have James killed, but the Romans refused on the grounds that James hadn't broken any laws. With the Romans now out of the picture for the time being, Ananus had James executed, along with other unnamed parties.

The people of Jerusalem seem to have been fond of James, probably because of his reputation for being committed to charity and collecting alms for the poor, widows, and orphans. They were also upset that Ananus had exceeded his lawful authority. When Albinus finally arrived in Jerusalem, the people complained about Ananus' actions during the absence of a proconsul. Albinus responded by deposing Ananus and appointing a man named Jesus son of Damneus (no relation to the Christian Jesus, of course).

But this younger Ananus, who, as we have told you already, took the high priesthood, was a bold man in his temper, and very insolent; he was also of the sect of the Sadducees, who are very rigid in judging offenders, above all the rest of the Jews, as we have already observed; when, therefore, Ananus was of this disposition, he thought he had now a proper opportunity. Festus was now dead, and Albinus was but upon the road; so he assembled the sanhedrin of judges, and brought before them the brother of Jesus who was called Christ, whose name was James, and some others; and when he had formed an accusation against them as breakers of the law, he delivered them to be stoned: but as for those who seemed the most equitable of the citizens, and such as were the most uneasy at the breach of the laws, they disliked what was done; they also sent to the king, desiring him to send to Ananus that he should act so no more, for that what he had already done was not to be justified; nay, some of them went also to meet Albinus, as he was upon his journey from Alexandria, and informed him that it was not lawful for Ananus to assemble a sanhedrin without his consent. Whereupon Albinus complied with what they said, and wrote in anger to Ananus, and threatened that he would bring him to punishment for what he had done; on which king Agrippa took the high priesthood from him, when he had ruled but three months, and made Jesus, the son of Damneus, high priest.
- Flavius Josephus: Antiquities of the Jews Book 20, Chapter 9, 1

Obviously, Jesus never makes any kind of claims about his divinity or messianic status, but we see something close to it - a Jewish writer making reference to the fact that Jesus "was called Christ". The specific use of the word "called" implies that Josephus doesn't necessarily share this assessment, but it does make it clear that he was aware that other people had called Jesus "Christ".