Dempsey scored 17 goals in 37 league appearances for Fulham this season and has made it clear that he would relish the chance to play with a bigger club in Europe's top competition.
That is a feeling that is shared by his national team coach, the former German international striker who is preparing his team for Saturday's friendly against Scotland.
"If you play consistently at the highest level then offers will come in. I think everybody is aware of that. His goal is the right goal, we have often discussed it," the former Inter Milan and Tottenham Hotspur striker said on a conference call.
"If he has the opportunity to go to a Champions League team, that's the creme de la creme of club soccer then you have to do that, with all the respect obviously to Fulham, as he often mentions.
"But you want to play in the biggest club competition in the world and that's the European Champions League and hopefully something will happen over the next couple of weeks," he said.
"If it doesn't happen then he has to keep pounding away and knocking at the door."
Klinsmann said he has not talked about Dempsey's club situation during training this week but said it had been a topic over the past season.
"We haven't talked about it yet [this week] - but before on the phone, following up his Premier League games we discussed different ways of approaching things," said Klinsmann.
Dempsey will not feature against Scotland in Jacksonville as he recovers from a groin injury but Klinsmann is hoping he will be back for Wednesday's friendly against Brazil as the Americans prepare for June's World Cup qualifying games.
Landon Donovan is in the U.S. squad after missing a number of friendly games during Klinsmann's first year in charge and Klinsmann sympathised with the player, who said this week that he was finding it hard to find the hunger for the game.
"I think every player goes through those moments. Some a bit earlier, some a bit later," said Klinsmann.
"I had those moments too, they came usually when you were kind of ready for the next step or a change. It helped me a lot to change [clubs] a few times more. It kept me on my toes and I kept learning a lot of things off the field too. I had those thoughts though," he said.
After facing Brazil in Baltimore, the U.S travel to Toronto to take on Canada on June 3.
The CONCACAF regional qualifiers for the World Cup begin on June 8 in Tampa when Klinsmann's team face Antigua and Barbuda and then four days later face Guatemala on the road.
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