The Argentine attacking midfielder has finalized his switch to White Hart Lane, with Gareth Bale's move to Real Madrid expected to follow as a result.
The Serie A outfit accept a bid said to be in the 30 million euro range for the attacker, who has signed a five-year contract with his new side. The deal sees Spurs break their transfer record for the third time this summer, following the acquisitions of Roberto Soldado and Paulinho.
Roma had confirmed the player's imminent sale on Wednesday, with Lamela flying to England to complete a physical. His addition bolsters Andre Villas-Boas' squad ahead of the expected confirmation of Gareth Bale's world-record transfer to Real Madrid.
Lamela becomes the club's second confirmed signing of Friday, with defender Vlad Chiriches having completed his move from Steaua Bucharest. Spurs are also expected to finalize the transfer of Danish playmaker Christian Eriksen after Ajax technical director Marc Overmars confirmed that the clubs are in negotiations.