Both of these sentences appear grammatically correct but the way I’d probably say it would be “There is a high chance that his intention is sincere.” or “There is a high chance that he had a sincere intent.”

The chance is high that he is sincere about his intentions. The chance is high that his intentions are pure. Chances are high that he has honest intentions. "His intentions" has a particular usage context as that of a man towards a woman, which "his intention" does not. It covers the general character of all future intentions towards her. In the case of a singular intention, we would be talking about a statement of intent about only one specific act.

They are correct but I am uncomfortable with the first. I have trouble imagining what it would mean for an intention not to be sincere. An intention can be good or bad, but an "insincere intention" would be an intention that is not really an intention at all. I think what the sentences are trying to say is that his apparent intentions are not really his true intentions. He does not really intend to do what he says he will do. A better way to say this would be "The chance is high that he is hiding his true intentions." or "His intentions are not as they seem."

Both sentences are grammatically correct, but the first one sounds more natural and idiomatic: "The chance is high that his intention is sincere." However, if you'd like to rephrase the second sentence to make it flow better, you could say: "There's a good chance he is sincere about his intention."