Verb for pressing upper and lower lips together?

There are some people that would call it to bite, but it is wrong because you don't use the teeth.

Is there a verb or idiom for pressing with the lips?

Something like this:

Solution 1:

This is called "pursed". In particular the second image illustrates this pretty well.

As some of the comments suggest, pursing can be ambiguous referring to outward/inward lip rolling. This following reference explains more, and explains the related body language signals (though, curiously, it all but omits the "pout").

Lip Body Language

"Pursed" lips are not to be confused with "pursed lip breathing", a technique used by folks suffering from COPD, et al, to achieve better air exchange in the lungs.

Wikipedia - Pursed lipped breathing

Solution 2:

I think the best term is sucked in.

Here is the relevant definition from Changing Minds' glossary on body language, specifically lip positioning:

Sucked in

When lips are sucked into the mouth or turned under so the red part of the lips are hidden ('swallowing the lips'), this can indicate that the person is thinking and uncertain about something, which could easily be bad news.