The 26-year-old has started 2012/13 in scintillating form, netting nine times in the league already this season, nearly matching last year's return of 11 goals at the halfway stage of the campaign.
Spurs are long-term admirers of the Argentine-born forward, who has been capped six times for the Italian national team, and it's believed that Andre Villas-Boas is still keen to bring him to White Hart Lane in the New Year.
With Jermain Defoe and Emmanuel Adebayor currently Tottenham's only striker options in north London, the Portuguese tactician has been tipped to strengthen his forward line next month.
The Premier League side will face a serious fight to secure Osvaldo's signature, though, with Italian publication La Nazione reporting that Fiorentina are preparing a bid of £12million.
Osvaldo came close to moving to England during the summer, but now looks set to stay in Italy. Roma remain reluctant to sell, but the possibility of helping Fiorentina's push for Champions League qualification might prove too good an opportunity for the player to turn down.