Brown had been out since picking up a separate knock in the warm-up to the Hoops 2-1 League Cup victory over Hearts at Tynecastle last month.
The Celtic captain, who was forced to sit out four matches, picked up a knock towards the end of the scoreless draw with Killie on Saturday and manager Ronny Deila is now resigned to losing the influential midfielder until well into the New Year.
Speaking ahead of Thursday's Europa League tussle with Ajax in Glasgow, Deila said Brown faces "eight to twelve weeks" out of action but he hopes surgery will not be required.
"Scott had a scan on Monday but we are not sure if he requires an operation," Deila said. "Hopefully it will not be necessary."
"It is always tough to lose your captain, but having said that we have done it before. Last year when he was out we won a lot of games.
"It is going to mean big changes in the midfield but we will put together a new one and we have a lot of good players"
The news is the latest blow for the Parkhead club ahead of their must-win Europa League clash with Ajax.