The 29-year-old has scored 16 goals for Fulham this season.
Dempsey’s current contract with the Cottagers is set to expire in the summer of 2013.
Fulham boss Martin Jol is keen for Dempsey to sign a new contract with the club, but the Texan could be interested in a move to a bigger club.
Roma are willing to pay £7million for the highly-rated USA international, according to The Sun.
Arsenal have also been linked with a summer move for Dempsey in recent months.
