The winger, 19, is also wanted by other Premier League and Championship clubs but Spurs have agreed a package worth around £100,000, which includes a sell-on fee of between 10 and 20 per cent for the Isthmian League Premier Division club.
Tanner has been training with Spurs’ development squad for the past fortnight, impressing coaches, but is yet to decide on his next club.
The teenager – formerly on the books at Arsenal and Charlton – picked up an injury in Tuesday’s 2-0 win over Cray Wanderers, which could complicate a move.
Spurs have a vacancy on the wing in their Under-23 side after selling Dilan Markanday to Championship promotion hopefuls Blackburn Rovers earlier this month.
Tanner has been watched by Spurs academy chief Chris Powell, having been recommended by Lewes boss Tony Russell.
Brighton are understood to have had an offer rejected that would have seen Tanner remain at Lewes on loan for the rest of the season.