The Ivory Coast star, who has scored four goals this season, suffered a knee injury during last weekend's 1-1 draw with Newcastle.
The news is a major blow to Palace's survival hopes, not least because they have not won a league game without him since September 2016.
There were fears the Eagles could be without their talisman for the rest of the season, but manager Roy Hodgson said: "The good news is he's a very quick healer.
"He and the medical staff are working very hard to get him back on the field as soon as possible.