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It is possible for him to be seeking closure but why the secrecy and denial? It is more hurtful to be lied to than talk about it and him let you know what is going on. That would be more reassuring.
Regardless of his intent (why he's communicating with her), he is with you now and has to respect you. Trust is important in any relationship and especially in a romantic one.
You won't want to snoop in on him even if you're suspicious or things lay out in the open (emails, texts, etc) but you also do not want him to give you the idea that he's not talking to her if in fact he is. If he had made the decision to talk to her, he has to be responsible enough to tell you about his decision. Then, the two of you can talk about its pros and cons. Couples usually resolve problems together.