It may well be useful to find out why. [From personal experience] There are things that can be done to mitigate bad breath, if it's confirmed.
I say this because, if it is true that you have bad breath, you can richly benefit by this information -- assuming you did not know this already. By the phrasing of your question, it looks like you are not sure.
May I suggest a first step to be whether you notice him doing this with others also. This may take some effort, but it may be worth it. If he does, then there may be nothing to bother over, as all you need to do is accept the gum only if you feel like having some. If he does not, then I suggest going to the next step.
How does he act if you politely say you are not in the mood for gum right now? If he makes his comments short and leaves, there may be cause for concern.
Going back to my first comment at the top, why not remove remaining doubts and visit a dentist. They usually have expertise and machinery that can lead to objective information given to you from a professional who has no reason to lie or spread discussions behind your back. If confirmed, he will be able to give professional advice on a plan of action that has the potential to significantly improve your quality of life. I state this as a last resort because it's usually a pricey option.