Press Association Sport understands the United States international is having a medical with the White Hart Lane club.
Aston Villa, Liverpool and Sunderland were interested in signing Dempsey, who was understood to prefer a switch to Merseyside. However, Tottenham have joined the race - and appear to have moved into pole position as the clock ticks down ahead of the 11pm transfer deadline.
Tottenham, who have also been linked with Shakhtar Donetsk's Willian, have money to spend following the £30million departure of Luka Modric and Rafael van der Vaart, who joined Hamburg for a fee reported to be around £10million.
Spurs were linked with a move for Portuguese midfielder Joao Moutinho but Andre Villas-Boas now appears to have his sights set on signing Dempsey before the deadline.
The club are also closing in on the signing of France goalkeeper Hugo Lloris.
Lloris arrived at Tottenham's training ground on Friday afternoon amid reports the north London club had agreed a £13million fee with the player's current club Lyon.
Press Association Sport understands Lloris has had a medical at the club and is on the brink of a move - despite furious claims made earlier by Lyon's chairman Jean-Michel Aulas about the conduct of his opposite number Daniel Levy in negotiations over the transfer.
Source: PA
Source: PA