A perennial question that some people raise is whether Jesus claimed to be God. The skepticism underlying the question is that while NT writers claim divinity for Jesus, that's not a claim he made for himself. In other words, he was just human, and a divine Jesus reflects legendary embellishment on the part of NT writers.
Now, in one respect, I think the question is unimportant. I mean, if Jesus is God Incarnate, then we'd expect him to indicate that fact, but what I mean is that Jesus didn't write anything, so if someone is skeptical about the historicity of the Gospels or the NT generally, they will be just as dismissive of accounts in which Jesus claims to be God. They will say the Gospel writers put those words on Jesus' lips. So when the question has that frame of reference, it's futile to distinguish what Jesus said about himself from what NT writers said about Jesus. Since we don't have an autobiography of Jesus, there's no point attempting to prove to a "skeptic" that Jesus claimed to be God. If they distrust the historicity of the Gospels, they'd say statements attributed to Jesus are reducible to what the Gospels authors said about him rather than what he said about himself. To that extent, I think a "quest for the historical Jesus" that labors to isolate his statements from the narrator's statements is pointless.
There is, though, a more interesting question. How would Jesus prove that he's divine? It's not enough to claim divinity. After all, some people claim to be God, but we typically dismiss them as crackpots.
So it's less about Jesus saying he was God than Jesus showing he was God. Mind you, saying that he was God would help to prep the observer, but that needs to be reinforced by corresponding actions. Doing things that are associated with divine action.
However, that, of itself, is not without ambiguities. For instance, God is not the only agent who can perform miracles.
Now normally, when a crackpot claims to be God, that doesn't pose a threat to the true religion since most folks don't take him seriously. Indeed, the claim itself is sufficient reason for them to discount him as either delusional or a charlatan.
If, however, a person made a credible claim to be God; if he garnered an enormous following; if, indeed, that became the dominant religion, then it would pose a threat to the true religion unless either the claimant is, indeed, what he claims to be, or else God intervenes to discredit him.
Take the cliche of the blasphemer who dares God to strike him dead. Normally, there's no lightning bolt that calls his bluff. But that's because the garden-variety blasphemer is not that important. God won't give him the satisfaction. God can't be compelled.
If, though, a religious impostor was so successful that he'd lead the faithful astray, then it's up to God to safeguard his name and to protect the faithful from mass deception and apostasy. The OT talks about how God is "jealous" about his name, which some readers might find a bit theatrical or egotistical, but it's in the context of heathen idolatry, where you had pagan religions holding humanity in their thrall.