Badstuber, who only returned 10 days ago after three weeks out with a muscle injury, turned his right knee after a challenge with fellow Germany international Mario Goetze during the 1-1 draw.
"Holger Badstuber has sustained a cruciate ligaments tear and is expected to be out of action for half a year," the Munich club in a statement.
Bayern, who lead the Bundesliga table by eight points, have also qualified for the knockout stage of the Champions League.
They host Belarussian side BATE Borisov next week in their last Group F match as they try to finish top.
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