The 27-year-old was stretchered off only minutes shy of the half-time whistle at the Imtech Arena in Hamburg, and it was later discovered that he would require an operation to rectify the problem.
Speaking on Sunday, Bayern sporting director Matthias Sammer admitted: "He is not good. He needs surgery. Then we'll see how long he's out for."
The match finished 2-2 after regular time, only for Thomas Schaaf's side to win out 4-2 on spot-kicks.
The Germany striker could join the likes of Rafinha (torn knee ligament), Diego Contento (hairline foot fracture) and David Alaba (stress foot fracture) on the club's treatment table with long-term injuries.