The Italians were keen to sign Fulham’s top scorer last summer.
And USA hotshot Dempsey, 29, who has hit 16 goals this term, is out of contract in June 2013.
Fulham boss Martin Jol is desperate for the Texan to commit to the club but Dempsey knows signing a new contract at Craven Cottage could stop him from trying his hand at one of Europe’s major clubs.
Arsenal have also been linked with a summer move but now Roma are ready to test Fulham’s resolve with a bid at the end of the season.
It is understood the Serie A giants are willing to pay £7million for the midfielder who could walk away for nothing in just 14 months.
