I tell them only if I’m seriously considering it or it’s beyond offensive (and they need a laugh).

I never mention other job offers. Once my employer overheard one and was pissed (not at me - at the woman).

Nanny here - generally, no. I get frequent inquiries and outright job offers all the time.

Nannies, do you tell your employer when you get another job offer or inquiry?

Employers, do your nannies tell you when someone has approached them about another job? Do you want them to tell you?