The 34-year-old appeared to jar his knee when he stood on top of the ball during the second half of his side's 2-2 Bundesliga draw away to Hertha Berlin on Sunday.
Tests have since revealed the extent of the damage and initial reports suggest he is facing up to three months out of action.
"We're sorry Franck has hurt himself," chairman Karl-Heinz Rummenigge said via the club's official website. "We all wish him a quick recovery."
Bayern said in a statement that Ribery has already begun rehabilitation work to repair the damage, although he will not require surgery.
Speaking after the match, the former Marseille star said: "I'm in a lot of pain. The outer ligament is probably damaged. I hope it's not any worse than that."
Team-mate Jerome Boateng tweeted soon after the news: "Get well soon bro!".