The Ivory Coast striker, a summer signing from Wigan, sustained what was first thought to be 'a knock' in the 2-1 win over Hull City.
But scans have revealed the damage to be more serious than expected and boss Roberto Martinez said it was inevitable Kone would be out for 'a long spell' after making only six appearances for his new club.
"We had disappointing news [by which Kone] will need a bit of a period away," Martinez said.
"We need to give him a lot of support because he was enjoying his role when he came on against Hull and then he got a nasty incident that is going to keep him away for a long time.
"He had a loosened body, a bit of cartilage in his knee, so we need to repair that which will take time.
"We're going to work on other aspects of his body to get him fully fit and strong so that when he comes back, he is ready to enjoy his football. It's going to be a long spell out and that's disappointing.
"We've got Darron Gibson reacting really well a couple of weeks after surgery so now he and Arouna are going to work extremely hard to get back before the season ends. We need to give them help and support on that side."