Kone was due to play his first senior game in a year when Everton played Swansea City in the League Cup in September, but felt pain in his knee during the warm-up and missed out.
"It is not a straightforward injury. He has been really unfortunate and it seems we have to be perfect in this final stage of the injury," Martinez is quoted as saying by the Daily Star.
"He's in pain after regular exercise and normally, at this stage of the injury, he should be okay after 24 hours. At the moment it's taking 48 hours to get back.
"I expect him to be fully fit by January and by that I mean match fit. He is going to be involved in 11-a-sides behind closed doors this week to build him up again and he will be fully fit in January."
Kone signed from Wigan for £6m in the summer of 2013.