The 20-year-old was injured at the French training camp in Clairefontaine while preparing for Friday's 2018 World Cup qualifier at the Stade de France against Sweden.
Bayern say a medical examination in Munich on Thursday revealed Coman has torn the cruciate ligament in his knee and ruptured the ankle capsule in his left leg. He is expected to be out for two months.
Coman will miss Bayern's six remaining league games of the year, plus their Champions League clashes at Russia's Rostov on 23 November and at home to Atletico Madrid on 6 December.
After sitting out the first two league games this season with an ankle injury, Coman has struggled to maintain the dynamic form that made him one of the stars of France's surge to the Euro 2016 final last summer. Since those heights, he has yet to play a full Bundesliga match and has managed only 25 minutes of Bayern's four Champions League games this season.
Bayern, who won three Bundesliga titles on the trot with Pep Guardiola before he left to take over at Manchester City, are top after 10 games with 24 points. Newly promoted RasenBallsport Leipzig also have 24 points but trail the Bavarians on goal difference.