Kayal was initially ruled out for at least one month last week after being injured in the Old Firm victory at Parkhead.
But manager Neil Lennon said: "He had a second scan and he needs surgery."
He added: "We don't know what the outcome of the surgery will be. There's a possibility that he could be out for four months. It would be a huge blow for me and the player and the club itself.
"It's no coincidence that this run has coincided with Beram signing a new contract. He is a pivotal player for us. But I will say one area we have real quality strength in depth is midfield. I've got plenty of cover in midfield.
"But no-one likes to lose a player for a concerted period of time. We have been unfortunate with these injuries."
Kayal suffered the injury during a challenge with Rangers midfielder Lee McCulloch.
"It was a good challenge between two committed players," said Lennon, whose team face Peterhead in the fourth round of the William Hill Scottish Cup fourth round on Sunday.
"The way his ankle has turned, he has got a bit of cartilage floating in there that might need attached back in and there is ligament damage as well.
"The operation is Monday. It could be a minimum four months or we could maybe get better news. The result is he needs surgery, we are just not sure on the extent of it yet."