The Italy international only joined Roma from Espanyol last season, but impressed in his debut season in Serie 'A', hitting 11 league goals and registering five assists.
And the 26-year-old has started this season in red-hot form, scoring five times in six Serie 'A' starts.
Roma boss Zdenek Zeman has proven to be something of a divisive character since joining the club this year, and the report suggest Osvaldo may have fallen out with the Czech coach.
Should that be the case, Juventus are ready to offer the Argentina-born striker an exit route, but Spurs are also thought to be considering a bid.
The £11m-rated forward has six caps for Italy - he only received his first call-up in 2011 - but already has three goals for his country.
Osvaldo qualifies for Italy through his great-grandfather, who was born in Italy and later emigrated to South America.
Tottenham are believed to be considering their options for the January transfer window, and Spurs boss Andre Villas-Boas reportedly has his eye on a number of strikers, including Osvaldo.
Tottenham face Wigan this weekend in the Premier League, and Villas-Boas has said he expects a tough test at White Hart Lane as Spurs look to bounce back after their midweek defeat to Norwich in the Capital One Cup.