The move would expand the 51-year-old actor's involvement in the project after previously entering talks to star in the big-budget movie opposite Margot Robbie.
According to Variety, Bradley is being considered to helm the film after Twisters director Lee Isaac Chung exited the project.
Lee had taken over directing duties after Jay Roach had also previously been attached to the film.
The crime heist prequel is being produced by Margot Robbie, through her production company LuckyChap and is intended to begin shooting before the end of the year if a director is finalised.
If Cooper formally took on directing duties, the film would mark his fourth feature behind the camera, following A Star Is Born, Maestro and the 2025 film Is This Thing On?
The new film is expected to be set before the events of the 2001 heist comedy Ocean's Eleven, which starred George Clooney, Matt Damon, Brad Pitt and Julia Roberts.
A screenplay draft for the prequel was written by Carrie Solomon, whose version of the story is reported to take place in Europe during the 1960s.
The Ocean's franchise itself began with the original 1960 film Ocean's 11, starring members of the Rat Pack including Frank Sinatra, Sammy Davis Jr, Joey Bishop, Peter Lawford and Dean Martin.
Decades later, director Steven Soderbergh revived the concept with the 2001 remake Ocean's Eleven.
The film later produced two sequels – Ocean's Twelve in 2004 and Ocean's Thirteen in 2007 – continuing the story of a group of sophisticated thieves planning elaborate heists.
The franchise was later revisited with the 2018 spin-off Ocean's Eight, which starred Sandra Bullock, Cate Blanchett, Anne Hathaway and Rihanna.
A separate new instalment in the franchise is also in development, focusing on the original group of characters as older criminals attempting another scheme.
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