Yes, if the auto-accept option has been selected. When you list an item with the Best Offer feature, you can opt to manually review all offers, or automatically accept or decline offers above or below a specific amount (you can't set a listing to automatically accept an offer without specifying a minimum amount, eg say you have an item for $50 and set it to accept any offer over $45 automatically).

If the auto-accept option has not been selected, any offers will expire after 48 hours.