According to the GI report, the service will offer first-party games and also allow third-party publishers to stream back catalogues.
The partnership is likely to be announced at E3 as part of Sony's conference on Monday.
Gaikai already carries portfolios from Electronic Arts, Ubisoft and Capcom, but all titles so far have been for PC.
Sony initially carried BC for PS3 from its launch in March 2009, but dropped support after the introduction of PS3 Slim back in August 2009.
Sony recently introduced PS2 Classics for the PlayStation Store for PS3 in the US and Europe. Some of the games as part of the program include Max Payne, Need for Speed: Most Wanted and SSX On Tour.
Users can also purchase PS1 Classics from the PS Store.