The 20-year-old will miss the rest of the season following the red-card offence during his team's 4-0 defeat last weekend.
"Valentin Eysseric, who was sent off at St Etienne last weekend, was suspended for 11 games by the French Professional League's (LFP) disciplinary committee. The player will therefore not play again this season," read a statement from Nice.
Clement, who required hospital treatment following the foul, will now undergo six months of rehabilitation after receiving surgery.
Eysseric has apologised for his actions and confirmed that he will not appeal against the disciplinary panel's decision.