The 29-year-old American netted 23 goals in all competitions this season to help Martin Jol's men finish ninth in the Barclays Premier League.
Dempsey - signed from New England Revolution in January 2007 - has been offered improved terms by the Cottagers.
However, in an interview with American magazine Sports Illustrated, he reiterated his desire to test himself against the game's best.
"I am grateful for everything Fulham have done for me, some of the best memories I have in soccer have been there, but the fact remains I want to play in Champions League. I want to play at the highest level possible," Dempsey said.
"I want to do as much as I can in my career and look back and say I made the most of all the opportunities I have had. I am always going to want to play in Champions League. It is something I have never done. I would like to test myself at that level. That way I can look back and say, 'Hey, I gave it a go. Either I was good enough or I wasn't good enough"'