This is a hard one. Your intuition is telling you something is not right. I had this same feeling. I knew the texts were coming from a woman at his work, but he just said it was about work, or about training (he trained her in cross fit). I got uncomfortable (that my intuition was nagging at me), so I asked him to stop spending time with her outside of work. He got all offended and acted like I was attacking him, like I did not trust him. I dropped it, feeling bad I had made him feel untrusted. A couple of months later this woman's husband called me and said he also suspected an affair. I called my husband and told him about the call, and he once again assured me that I was his only one, and that he would investigate as to why this man would suspect such a thing. 2 hours later the man called me back, and told me that after hours of fighting, his wife had finally admitted the affair with my husband. I remained calm, called my husband and told him I was not calling to talk, I was simply calling to inform him that I knew about the affair, that she had admitted it, and that I was leaving him. His response was "I am sorry".
If you ask him about this without proof, he will more than likely say he is not cheating, even if he is. You do need to put this to rest for your own sanity. Start by talking with him, but if he denies and you still feel something is just not right, than do what you need to do to find the truth.
Blessed Be