The Dutch international had to be taken off in the 24th minute of their second-round match.
"Matthijs de Ligt sustained a partial tear of the ligament in his right knee joint," the German champions said in a statement.
"That's the outcome of a scan carried out by the club's medical department on Thursday. Centre-back de Ligt will therefore miss FC Bayern's upcoming games."
The Bavarians play title rivals Borussia Dortmund on Saturday and then host Turkey's Galatasaray in the Champions League on Wednesday.
Bayern, who are already missing injured centre-back Dayot Upamecano and will also be without suspended defensive midfielder Joshua Kimmich against Dortmund, are currently in second place in the Bundesliga two points behind leaders Bayer Leverkusen.