The 29-year-old, who will undergo further medical tests, suffered the blow to his knee during the Merseyside outfit's 2-1 victory over Hull City last month.
"We had disappointing news [by which Kone] will need a bit of a period away," Martinez told the club's official website. "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."