The Cottagers were not willing to budge on their £12m price-tag for the American star. However, according to Metro, the Black Cats are willing to pay Fulham's asking price of £8m for the 29-year-old who scored 23 goals in 46 appearances last season.
Fulham earlier reported to the FA regarding Liverpool's interest, saying Dempsey had been 'tapped up'. The 29-year-old refused to play and hence was ignored for the opening two games.
Fulham manager Martin Jol had said, "It would have been better if Liverpool followed up with a formal bid, but they didn't, so until now there is no bid."
Liverpool were ready to offload midfielder Charlie Adam, who has lost his place in the starting XI with the arrival of Nuri Sahin and Joe Allen, as an exchange offer plus cash to the Londoners while defensive midfielder Jay Spearing is set to join Championship side Bolton Wanderers.