The Anna Karenina director is thought to be the favourite to lead the erotic novel's screen adaptation, according to Variety.
Universal and Focus Features, who are making the film of EL James' best seller, haven't confirmed Joe has been hired yet but the Hollywood rumour is that the job is his.
The search has been on for a while to find a director who could make the "mummy porn" tale look classy.
Other names in the hat were said to include Monster director Patty Jenkins, Twilight: Breaking Dawn's Bill Condon, Moneyball's Bennett Miller and Good Will Hunting director Gus Van Sant.
Steven Soderbergh (Ocean's Eleven) was said to be in the running, too.
Writer Kelly Marcel is still working on the script and the producers have been confirmed as The Social Network team Michael De Luca and Dana Brunetti, leaving the way open for casting characters Christian Grey and Anastasia Steele to begin.
Last month, The Wrap reported that Van Sant was so keen to get the directing job he had even shot a scene to send in as a test tape for the job. He is said to have cast Magic Mike star Alex Pettyfer in the role of Christian Grey.
50 Shades Of Grey by EL James became the best-selling book of all time in Britain last year, overtaking The Highway Code and Harry Potter.
Could Alex Pettyfer play Christian Grey?