Does it sound like he likes her? She likes him?

First things first:
I love my fiancé, I really, really do.
She's the best thing that's ever happened to me and I don't want to lose her.
But at the same time we've lived in this new (if it can even be called that anymore) city for 3 years and I want her to have friends but something about this seems off to me and I'm looking for other opinions.

The Story: As I mentioned we moved here 3 years ago and haven't really made any real friends.
Now, my fiancé is a teacher and there's this new sub that's been around her school since Christmas that she's mentioned a few times.

I know in the past she's always gotten along more with guys than girls and because of that I really don't get jealous about much. The thing is that we've lived here for 3 years now and she hasn't made any friends and I feel bad for being jealous this time, as I do trust her, but for some reason weary this time.

So, here's where we are.

She's becoming friends with this guy at work.

She doesn't want to get too close as she thinks "I might get jealous" and she says she "doesn't want him or me to get the wrong idea." She says she makes it a point to mention me when chatting to this guy, who she reassures me, does not have any interest in her.

But the thing is, I feel that he does...or she subconsciously (Or not) does.

She asks him questions about his interests/hobbies, which are basically all the same as mine and things that I introduced her to....which I find a bit weird. Now, it be just me, but by asking about someone's hobbies and interests are you not conveying interest in them and possibly sending a mixed message?

But she never mentions that he asks her questions or me...or that they talk about her outside of school...or me for that matter other than the "Mentioning me"

He's made jokes about not having seen her at the gym. He even invited her once, but she said "No" as "She wouldn't know what she was doing" and when I offered to come with her she didn't seem interested at all and I'm a kinesiologist/Trainer. Maybe she wants to seem like she knows what she's doing around someone else?


Does it sound like
- He possibly likes her?
- She possibly likes him"
- That it's a harmless relationship?
- Should I suggest meeting this guy? How would I bring that up without coming off as a jealous ass

And please explain your answer as I'm really scared that the fiancé is on the way into another guys arms and I don't know what to do about it.

Is there anything I can do if it is a crush??