Robben picked up the knee injury in the DFB-Pokal win over Augsburg on Wednesday night.
"Arjen has a deep cut, right into his knee joint," said club doctor Hans-Wilhelm Muller-Wohlfahrt following tests on Thursday morning.
"A stud put a lot of pressure on the femur in his right leg and there is a big build-up of blood."
The injury will see Robben miss the Club World Cup in Morocco.
"Of course I'm bitterly disappointed," Robben said on Bayern's website. "I was in such good form and was looking forward to every game I was playing, and now this happens.
"But I'll come back fit and will then help the team reach the goals that we set ourselves before the start of the season."
"We're incredibly sorry that Arjen got injured so badly in Augsburg," said Bayern's director of sport Matthias Sammer.
"Arjen was probably in the form of his life right now and been decisive in so many matches so to lose him now is bitter, but he will get all the help he needs from us and will be back for the second half of the season."