1. May I use your telephone?
2. May I borrow your telephone?
#1 is supposed to be correct. Is #2 also acceptable?
I don't know if it happens in AmE, but in BrE we have been known to use "borrow" for some rather silly things. For example:
Can I borrow your toilet?
This simply means "Please can I use your toilet/loo/bathroom etc?" Obviously, we don't actually literally want to borrow the toilet. We're not going to remove it from the bathroom, take it home and then bring it back later!