The Serie 'A' side have reportedly approached the Gunners over a possible fee for the Czech international, who is yet to feature for Arsenal this season after another injury setback.
The former-Borussia Dortmund playmaker has been plagued by injury troubles since he signed for the Gunners in 2006, but Roma are thought to be keen to bring him to Italy.
The 32-year-old sustained an Achilles' tendon injury last summer, and the report suggests Arsenal boss Arsene Wenger could be willing to cut his losses and cash in on the midfielder when the transfer window reopens in January.
Rosicky's Arsenal contract expires in 2014, and Roma believe they can secure a cut-price deal for the experienced midfielder.
The Gunners have plenty of options in centre midfield, following the return to the first-team of long-term injury casualty Jack Wilshere.
Arsene Wenger can count on Wilshere, Aaron Ramsey, Mikel Arteta, Santi Cazorla, Francis Coquelin, and the Arsenal boss recently sent Emmanuel Frimpong on loan to Championship side Charlton Athletic, so Rosicky may be deemed surplus to requirements come the New Year.