The Ivory Coast international sustained the problem in the 2-1 win over Hull on October 19 and missed last week's win at Aston Villa.
He is set to miss Sunday's visit of Tottenham but Martinez does not yet know how long the forward, whose Everton career has yet to take off after following his manager from Wigan in the summer, will be out for.
"We had a little bit of a setback and we are a bit worried with the knock he picked up against Hull," said the Toffees boss.
"I am a bit concerned with his knee because he is someone who gets through anything.
"Little niggles will never stop him from playing or being available and I have never seen him like this. We need to make sure we get to the root of the problem.
"We are going to assess him again and see because it is a little bit more serious than expected.
"He has over-extended the knee. It is something we need to find out about over the next couple of days.
"We have to do a couple more tests to see how the knee is reacting."
Kone's former Wigan team-mate Antolin Alcaraz, who has yet to make his Everton debut after being sidelined by a hernia problem since signing in the summer, will be available for selection later this month.
"Antolin is progressing really well but he is not going to be ready until the next international break," added Martinez.
Source: PA
Source: PA