The 22-year-old midfielder suffered ruptured cruciate ligaments during Friday's 2-2 draw against modest Caen.
The injury "will require surgery in the coming days," Nice said in a statement.
Cyprien's injury follows that of Nice's top scorer Alassane Plea whose season ended last month after a picking up a serious knee injury.
Cyprien played 29 times in Ligue 1 this season, scoring eight goals with three decisive passes.
Nice are currently provisionally second in Ligue I two points behind Monaco with champions Paris Saint-Germain a point behind.
Monaco host Bordeaux on Saturday with PSG, hurting after their sensational midweek Champions League implosion against Barcelona, at Lorient on Sunday.