The 19-year-old winger has been training with Spurs’ development squad for the past fortnight, impressing coaches, while continuing to play for the Isthmian Premier Division club.
Lewes are due to hold a board meeting on Thursday to discuss Spurs’ bid, with the Premier League club not yet thought to have met the teenager’s asking price.
There is widespread interest in Tanner from a host of other professional clubs, including Brighton.
The Seagulls are understood to have had a bid rejected that would have seen Tanner remain at Lewes on loan for the rest of the season, an option which has not been proposed by Spurs.
Lewes boss Tony Russell recommended Tanner to Spurs academy chief Chris Powell, who has been at the Dripping Pan, Lewes’ home ground, to watch the forward for their last two home matches. Tanner is in the squad for tonight’s home game against Cray Wanderers.
Spurs have a vacancy on the wing in their U-23 squad after selling Dilan Markanday to Blackburn Rovers last week.
Tanner, who is a quick and technically gifted player with an eye for goal, had spells at Arsenal and Charlton as a schoolboy.