Robben was taken off the field on a stretcher after he was hurt in a tackle during the first half of Wednesday's 2-0 victory at Augsburg.
"I was in such a good shape, I was looking forward to every game and now this," Robben said.
Robben had a deep laceration, club doctor Hans-Wilhelm Mueller-Wohlfahrt said after the Netherlands winger was examined at an Augsburg hospital.
Sporting director Matthias Sammer said he was "endlessly sorry" about Robben's injury.
"Arjen was in the form of his life and has helped decide so many games for us recently," Sammer said.
Robben scored five goals in the last four games. He scored Bayern's first goal in Augsburg, before being taken out on a stretcher in the 16th minute.
Augsburg goalkeeper Marwin Hitz was responsible for Robben's injury. Robben was called for offside but continued running with the ball and Hitz charged off his line to bring Robben down with a hard tackle.
Hitz said he challenged Robben because he didn't know if the winger would stop.
"I was late and really hit him. I am very sorry," Hitz said.
Robben will miss the Club World Cup in Morocco later this month, next week's Champions League match against Manchester City and Bundesliga games against Werder Bremen and Hamburger SV.
The good news for Bayern is that Franck Ribery is back after cracking a rib in a match for France on Nov. 15.