Can I ask for an extension on the time to accept a job offer?
Or will it put my chances for the position in jeopardy?


The position has been offered to you. The typical advice is the worst thing that can happen is they say no. There are some horror stories were an offer was revoked, but I would suggest you question wanting to work in any department that behaves that way.

It is completely reasonable to ask for a week or two week extension. When a school moves fast between visit and offer, it is unlikely you will get more than a week or two extension. That said, it cannot hurt to ask.