The 26-year-old, who earned 24 All Blacks caps, had an operation on Monday, according to a report in L'Equipe.
Fekitoa is not expected to play again for his Toulon club until late February or March.
Tonga-born Fekitoa joined Toulon in 2017 after 65 games for the Highlanders.
He has scored eight tries in 30 matches for Toulon and has appeared in 10 Top 14 games and four European Champions Cup fixtures this season.
Fekitoa's absence is compounded by Toulon's expected loss of French test centre Mathieu Bastareaud for the Six Nations tournament in February and March.
Former France test fullback Hugo Bonneval is also out until February and young lock Swan Rebbadji, who had ankle surgery in December 2017, is to have another operation this week.
Fekitoa's injury could lead to an imminent return to the Toulon starting team for his former All Blacks teammate Julian Savea, who has not played since before Christmas after taking leave to attend All Black brother Ardie Savea's wedding in Fiji.
Fekitoa and Savea are among 10 members of the All Blacks' 2015 Rugby World Cup winning squad now playing in France.
They are set to be joined at Toulon by another, wing Nehe Milner-Skudder, after the 2019 Rugby World Cup.
Former All Blacks and Chiefs back rower Liam Messam is also on Toulon's books.