So I find this a little confusing and it sounds like you’re in a very gossipy environment if you’re hearing from others what this woman is saying about your boyfriend. I’m assuming you’re in HS, but then again, my grad school experience may as well have been HS in terms of gossip. I’m also curious how you know what they talk about. Like, is he open about it with you, are these more things you’re hearing through the rumor mill, are you going through his phone?
Anyway, at any age, I’d not love a boyfriend being in what sounds like constant contact (over text/social media is what I’m assuming?) with a woman he used to have a “thing” with/for when we were getting together and for him to be talking about our relationship with her. If that crosses a boundary for you, that’s valid. It sounds like you’ve told him the gossip you’ve heard, but have you expressed your discomfort? If not, you can and should do so. Even how he reacts to you telling him you’re uncomfortable with the behavior will tell you a lot IMO.