The USA international scored 23 times in all competitions for the Cottagers this season and has been attracting interest from a number of clubs.
Dempsey has one year left on his current deal with Martin Jol’s side, and the striker has revealed an improved offer has been made, but his aspirations for elite European football will put several clubs on alert.
"The fact remains that I want to play in Champions League. I want to play at the highest level possible" - Clint Dempsey
"I've always said I'm grateful for everything Fulham has 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, 29, signed for Fulham in 2007 and admits that if the chance came to play at the highest club level, he would regret turning it down.
"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've had," he added. "I'm always going to want to play in Champions League. It's something I've never done.
"I'd 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’."
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