I've heard both "ay ess ay pee" and "ay-sap." It's not too common. My advice would be NOT to try to pronounce it at all, but just say "as soon as possible" or "right away."

This is a good American style pronunciation: https://www.youtube.com/watch?v=_vaUic34x5E

You just sound out the letters separately. So A-S-A-P. Some people will also make it a word and say 'assap'.

Thanks a lot for all your answers! It's very helpful!

Ay-ssap.