According to Italian newspaper Corriere dello Sport, the 27-year-old striker will be sold when the transfer window reopens, following a public falling out, during Sunday's 3-1 loss to Sampdoria in Serie A.
With Roma a goal down they were awarded a penalty, but Osvaldo refused to let Totti - the side's regular set-piece specialist - have the ball.
And, to make matters worse, the Argentine-born Italian then missed his spot-kick and was forced to make an apology via Twitter on Monday, saying he meant no disrespect to Totti.
Spurs boss Andre Villas-Boas has been credited with an interest in signing Osvaldo for some time, and he came close to moving to White Hart Lane last month.
Now, Tottenham's hopes of securing the former Espanyol forward's signature at the end of the season have been strengthened, with growing rumours of unrest in Rome.
It's unclear how much money Osvaldo will cost the traditionally frugal north London club, but after arriving at Stadio Olimpico in a reported €17.5 million deal, a Premier League switch is unlikely to come cheap.