The 27-year-old was a key player for Fulham last season and featured for the USA at the World Cup finals this summer.
“We don’t have an offer from Napoli,” Dempsey’s agent Lyle Yorks told the Major League Soccer website.
“There is a lot of interest from big clubs in Europe in Clint. And now you’ve got Roy Hodgson leaving for Liverpool… so there’s a lot of different things going on.”
He added: “We’ve got in Italy a major club and in France a major club. And in the Bundesliga a club. We’ll be sorting through them in the coming weeks and also seeing what happens at Fulham.
“There’s direct contact and discussions, but first they’d have to agree with Fulham.”
