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.
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 completed an 11million euro move from Fiorentina.
Lamela believes he could have done more to help the capital club improve on consecutive seventh and sixth-placed finishes during his two seasons.
He told the Gazzetta dello Sport: "I want to thank and bid farewell to the fans who always treated me well. I embrace them all and I'll always keep them in my heart.
"I'm proud to have played for Roma and I hope to come back one day.
"Maybe I didn't manage to achieve everything I wanted to, or could have.
"It's also true that we were lacking something as a team but I hope it goes better this year. I can see that all the lads have the desire to well and they're very focused.
"I hope the club, coach and my team-mates can go as far as they possibly can.
"For me, I hope to do well in London with Tottenham - and I'm ready."