The 27-year-old Colombia international sustained the injury in training on Tuesday, the club said, and will now need at least 10 days before being able to start therapy and his recovery.
"Bayern will have to make do without James Rodriguez for the coming weeks," it said in a brief statement.
The injury could not have come at a worse time for Bayern, who have dropped to fifth in the Bundesliga following their 3-2 loss to leaders Borussia Dortmund last week.
The Bavarians are already without injured midfielders Corentin Tolisso and Kingsley Coman. The Bundesliga resumes next week after an international break.
(Reporting by Karolos Grohmann; Editing by Jan Harvey)
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