may I offer you (something)?

This is a very formal and polite way to offer something to someone. Waiters, flight attendants, and other people in service jobs speak this way:

May I offer you today's newspaper?

May I offer you a dessert menu?

You can also use this phrase if you have guests in your home or office who you don't know very well:

May I offer you something to drink?

This phrase appears in these lessons: