The 27-year-old has been heavily linked with a move away from the Stadio Olimpico this summer, with Italian rivals Juventus credited with an interest in his signature.
Russian side Anzhi Makhachkala are also believed to be monitoring the Argentinian-born Italy international's situation, ahead of making a rival bid.
With 11 goals in 22 Serie 'A' appearances this season, the South American would add some much-needed firepower to Tottenham's forward line, with Jermain Defoe and Emmanuel Adebayor the only recognised strikers currently plying their trades in north London.
Osvaldo still has three years to run on his contract with the Giallorossi, but after a recent falling out with club legend Francesco Totti, head coach Aurelio Andreazzoli might be forced to sell the player.
Tottenham will hope to capitalise on Roma's transfer intentions, but know they will face a serious fight for his signature, when the transfer window reopens on July 1.