The 40-year-old director is set to helm the big screen adaptation of EL James' erotic novel, which will be produced by Universal and Focus Features, according to The Hollywood Reporter.
The two studios are reportedly looking for a director with 'arthouse sensibilities' for the flick and believe Wright could be the perfect fit for the job, although no offer has been made yet.
It is thought the director's long history with Focus Features, as well as his success working with young female protagonists - particularly long time collaborator Keira Knightley - could help him land the directorial role.
Other directors on the studios' short list include Gus Van Sant - who filmed a raunchy demo with 'Magic Mike' star Alex Pettyfer as lead character Christian Grey in a bid to land the job - as well as 'Monster' director Patty Jenkins and Bennett Miller.
'Fifty Shades of Grey' will be produced by Michael De Luca and Dana Brunetti.
The best-selling book trilogy tells the story of a young university student, Anastasia Steele, who embarks on a steamy relationship with her bondage-loving boss, Christian Grey.