The Daily Star claimed that the West London club will make a move for the Argentine-born Italy striker, who has been angling for a transfer away from the Giallorossi.
Other newspapers added weight to the claims, as the Whites look for a partner to Bulgarian star Dimitar Berbatov, following the departure of Mladen Petric.
Despite wanting to leave Stadio Olimpico though, the 27-year-old Osvaldo has already turned down moves to Southampton and Zenit St Petersburg.
However, Martin Jol’s men are hoping to lure the former Espanyol forward to Craven Cottage with the promise of regular Premier League football.
Osvaldo started his professional career at Huracan, before moving to Italy with Atalanta.
After spells at Lecce, Fiorentina and Bologna, Osvaldo came into his own at Spanish Liga side Espanyol, before returning to Italy with Roma in 2011.
He has scored 27 times in 55 Serie A matches for Roma, was last season's top scorer at the club with 16 league goals, and has bagged three in eight appearances for the Azzurri.
While his talent is unquestionable, Osvaldo has a reputation for being temperamental, falling out with the club after he was benched for the Coppa Italia defeat to Lazio.
A deal would represent something of a coup for Fulham, but the two clubs are likely to come to an agreement after cordial negotiations saw Maarten Stekelenburg make the same move this summer.