The Reds have been long-term admirers of the American, who has apparently refused to play for the Cottagers in a bid to force through a switch to Anfield.
A host of other Premier League clubs including Sunderland and Villa have been credited with an interest, with the latter having a £7m bid accepted by Fulham earlier today.
However, Dempsey has rejected the chance to link up with Paul Lambert at Villa Park. Liverpool have yet to make an official approach for the 29-year-old.
After failing in their Dempsey pursuit, Villa have switched their attention to 21-year-old Genk striker Christian Benteke.