According to the Daily Mail, David Moyes is ready to test the Cottagers' resolve with a £5m bid this summer.
The 29-year-old, who has scored 22 goals this season, has also caught the eye of Arsenal and Tottenham Hotspur.
Dempsey held talks with the Toffees in 2007 before making the move to Craven Cottage.
Moyes is said to have asked the American to wait until he had more funds available and money could be a problem for the Merseyside club once again this summer.
Fulham are believed to want somewhere around the £8m mark for the goal-scoring midfielder, who has one year left on his current contract.