Would he mention a girl he's hitting on?

I've been hanging out with this guy, and we had a little bit of a disagreement, and our relationship must be kept a secret at work.

He mentioned that this girl at work asked if he was texting me. And I asked why would she ask that, and he said he didn't know. And showed me texts she sent him - they were about work and something she had lost, and then asked somebody else to find it. He said she asked that out of the blue.

Would he tell me about her and her question if he or she was hitting on eachother?

Would he mention a girl he's hitting on?
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