Reports in Germany claim Spurs have had an £8million bid accepted by Hoffenheim for the Iceland international.
Swansea were the first club to agree terms with the German club but after manager Brendan Rodgers' move to Liverpool it appeared the 22-year-old would follow the Northern Irishman - under whom he excelled in a five-month loan spell from last January - to Anfield.
However, Press Association Sport understands that although the player is keen to be reunited with Rodgers discussions have stalled due to Sigurdsson's wage demands.
Tottenham, therefore, have emerged as an alternative destination for the former Reading midfielder who scored seven goals in 19 matches for Swansea last season and was named Premier League player of the month for March.
Hoffenheim have given Sigurdsson until Saturday to secure a move to England having allowed him extended leave after they returned to pre-season training just over a week ago.
Swansea thought they had secured the player on a permanent basis after agreeing a club-record £6.8million fee soon after the end of the season but before Sigurdsson could complete a medical their manager was lured to Liverpool.
Rodgers admitted he would be interested in signing the midfielder for the Reds but stressed the player had to first speak to new Swansea boss Michael Laudrup.
When the prospect of him moving to the Liberty Stadium having diminished it seemed Anfield would be his next stopping place but now it could be White Hart Lane.
Last week Hoffenheim coach Markus Babbel, himself a former Liverpool player, said of Sigurdsson's planned move: "If, by the deadline on June 30, he has done nothing then he will return to his pre-season with us.
"We have agreed this with him."
Sigurdsson is contracted to Hoffenheim for another two years but despite impressing in his first campaign in 2010/11 he dropped down the pecking order last season, forcing his loan to Swansea.