The Argentine youngster has been heavily linked with a switch to Spurs this summer as a replacement for the Real Madrid-bound Gareth Bale.
The 21-year-old scored 15 goals in 33 Serie A matches last season, also providing five assists.
Lamela is known for his exceptional pace and dribbling ability and much like Bale can play either on the wing or as a secondary forward.
He joined Roma from River Plate in August 2011 for fee thought to be €12m, rising to €20m
He has been capped three times by Argentina.
Having sold Lamela to Spurs, Roma then confirmed the arrival of Adem Ljajic from Fiorentina for €11m + €4m in bonuses.
Roma confirmed both deals in statements to the Italian Stock Exchange.