The American international is currently playing in his homeland for the Seattle Sounders after leaving Spurs over the summer.
With the MLS set for its off-season over the winter, players are free to complete short-team switches to European sides, so long as they have their parent club’s permission.
This has alerted the quartet, who are keen to bring the 30-year-old back to England.
The Sounders are said to be willing to let Dempsey leave on a temporary basis to keep his fitness levels up ahead of the 2014 World Cup, and the player himself is also believed to be open to the possibility.
Tottenham lead the race after inserting a first refusal clause into the deal that took him Stateside from White Hart Lane during the last transfer window.
But, Fulham are also looking at bringing back their former idol, while Aston Villa and Everton – both of whom have brought in MLS players on loan in recent years – are also in the hunt.
Seattle’s general manager, Adrian Hanauer, says that a deal is likely to happen.
“Clint’s contract allows him to go on loan for the off-season,” Hanauer said.
“All parties are working on a deal and we won’t be surprised if it happens.”
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