The 21-year-old has scored 19 league goals in two seasons with the Italian capital club and, should his move to White Hart Lane go through, he will help fill the gap left by Gareth Bale should he complete a world-record £86m move to Real Madrid.
A statement published on Roma's official website read: "Roma announce they have signed a contract with Tottenham for the outright sale of the playing rights of Erik Lamela, in exchange for 30million euros.
"The agreement also includes the obligation to pay 5million euros in bonuses related to the achievements of the player and the English club."
Lamela, who has been capped three times for Argentina, arrived at the Stadio Olimpico from River Plate in August 2011 and made 29 appearances, scoring four goals, in his first year in Italy.
“I am proud to have played for Roma and hope one day to return,” he told news agency ANSA.
"Perhaps I wasn’t able to express all I could’ve and wanted to do here. It’s also true that there was something missing for the whole team.
"Now I hope to do well with Tottenham in London. I am ready. I want to thank and salute the fans who always behaved well with me. I embrace them all. They will always be in my heart.”
A run of seven goals in six games at the start of the 2012-13 season saw him become an essential part of Roma's forward line, even though the campaign ended with a disappointing sixth-place finish and defeat to city rivals Lazio in the Coppa Italia final.
With Wales winger Bale absent from training as a dispute over a move to Real Madrid rumbles on, and with Spurs missing out on Willian following the Brazilian playmaker's decision to join Chelsea instead, the pacey Lamela offers a viable and talented alternative to both players.
The continued spending in north London indicates Bale's £86m move to the Spanish capital is all but ratified as Lamela joins Roberto Soldado, Paulinho, Nacer Chadli and Etienne Capoue in Andre Villas-Boas' new-look squad.
Lamela's place behind Francesco Totti in the Giallorossi team, meanwhile, will be filled by Serbian forward Adem Ljajic, who has today completed an 11million euro move from Fiorentina.