Brown sprained his right ankle, on which he had two operations in 2009, during Scotland's win over Denmark in August and aggravated the injury in the recent Old Firm defeat.
Celtic boss Neil Lennon revealed: "He's going to see a specialist again in the next few days. He will be out on Sunday too.
"I don't think it will require surgery but it might just be that it will take a bit longer than we first thought.
"We're hoping it's not as severe as a few months, but it might be a bit longer than a couple of weeks."