You're offered a chance to visit Island 1 by Bill after beating the Cinnabar Gym. And I think the other offer is made after beating the Elite 4 by Bill if you did not accept it earlier (not sure about this part). Check the FAQs for more information regarding the Islands.
"Hell, I'm surprised that you didn't ask 'should I breathe?' Life really is the most complicated of all games" - gbchaosmaster