From your recent post, it's clear he fancies her.
He's being totally unprofessional and this is beyond banter. It seems like she realises he likes her and has declined going on the trip for this reason.
As a HR professional, I'd actually say it's bordering on sexual harassment from him.
She's his subordinate and he's talking about making it worth her while to go on the trip.... That's way out of line.
There's a couple of ways to deal with it... Or that I might deal with it.
I'd confront him about the emails, under the guise of having received an anonymous call or message about the two of them, which prompted me to check the messages.
The thing is I don't like to reveal my sources and I fear that he'll just hide it if you don't say anything.
I had was told this next idea was passive aggressive, so I didn't actually do it, but I was going buy a copy of the book 'not just friends' and give my DH to read, when I discovered he was in touch with his Ex.
It was suggested that I talk through things, but I did send him an excerpt of the book to make my point, then we discussed it
Right now, it's not an affair, but he certainly wants more from her and if he doesn't, he's putting himself in a difficult position, because if an employee came to HR with his messages, we'd be having words with him and potentially taking action short of disciplinary.
I know I'd feel uncomfortable receiving that message from my manager.
Do you have children with him?
You could ask to see the messages between them (in his presence) to keep my mind at rest, because (and this will shake him up), you that I was