The 29-year-old American netted 23 goals in all competitions last 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, the United States forward 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 that 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'"
Dempsey continued: "If I wanted to be comfortable, I would have stayed in the MLS [Major League Soccer], because I would have been closer to my family and friends, and I would have been used to the way of life.
"I am a risk-taker. I want to throw myself in the deep end and see if I can swim."
Both Arsenal and Paris St Germain are said to be tracking the American.
However, Dempsey will not let himself get distracted.
Speaking to FOX Soccer, the United States striker added: "I don't get caught up in that [speculation] because until somebody is putting down some money it doesn't matter.
"If teams are really interested, they need to put their money with their mouth is and make an offer."
Dempsey missed Fulham's final game of the Premier League season with a groin strain and is therefore a doubt for the USA's friendly with Scotland on Saturday.
Manager Martin Jol believes Fulham can improve on their ninth-placed finish if he can keep his best players, with Dempsey among that contingent.
He told the Fulham Chronicle: "If all the players are still there, I believe we can do even better.
"I think playing in Europe cost us a few points, that is for sure.
"I can imagine Stoke played as many games as us and they struggled a bit and it's not a surprise that we were in the top five over the last 20 games, because we weren't involved in Europe."