The winger had to be helped off after an hour of his side's 2-1 win over Hoffenheim in the Bundesliga on Tuesday after falling awkwardly. An MRI scan in Munich on Wednesday revealed he has torn ligaments in his right ankle.
Ribery, who missed five months of football last season with a knee injury, will miss Bayern's next two Champions League games, including its match against Basel next Tuesday.
"I had feared worse," said Bayern coach Louis van Gaal after hearing the news. "Our misfortune is a blessing in disguise. Now we can look to getting Franck cured properly."
Ribery is expected to be in a plaster cast for the next 10 days, then will wear a special cast for a further 10 before he can begin light training.
If his recovery goes according to plan, Ribery is expected to return to team training in the second half of October.
Ribery's ankle injury happened during Bayern's 2-1 win at Hoffenheim.
Having been stunned by a goal after just 36 seconds, Bayern hit back as Ribery's fierce long-range shot was followed up by Thomas Müller on 62 minutes before Munich defender Daniel van Buyten's winner in added time.
Bayern got off to the worst possible start when they conceded the fastest goal in Hoffenheim's history.
The defending champions were stunned as midfielder Sebastian Rudy beat Munich's defence on the right and squared for Hoffenheim's Veded Ibisevic to score.
Daily Bild reported in its Wednesday edition that Van Gaal was set to sign a year's extension to his contract to stay with Bayern until 2012.
Having coached Bayern to the German league and cup titles last season, van Gaal is in his second season with Munich.
"Nothing is signed. All I can say is that we are very happy with the work of Louis van Gaal," said Bayern's sporting director Christian Nerlinger.
The result in Hoffenheim lifts Bayern up to seventh while the hosts are second.
Meanwhile, Bundesliga leaders Mainz 05 beat Cologne 2-0 on Tuesday to stay on top. The club picked up its fifth straight win to keep its 100 percent record and open a five-point lead.
The hosts took the lead when substitute midfielder Lewis Holtby's header beat Cologne goalkeeper Faryd Mondragon on 72 minutes before he scored his second in added time.
Champions League side Werder Bremen slumped to their second consecutive league defeat when they were beaten 4-1 at Hanover.
Bremen faces reigning champions Inter Milan next Wednesday in Italy, but Thomas Schaaf's side is lacking confidence after following up Saturday's defeat at home to Mainz with this loss in Hanover.
An own-goal by Germany defender Clemens Fritz after just 10 minutes put Bremen on the back foot until captain Torsten Frings converted a penalty after 36 minutes to put his side level.
Hanover reclaimed the lead when Ivory Coast striker Didier Ya Konan scored in the 53rd minute before defender Christian Schulz scored the second on 79 minutes.
To add insult in extra time, Hanover's Norway striker Mohammed Abdellaoue added the fourth to put his side third and Bremen 12th.
In Wednesday's games, bottom side Schalke are at fourth-placed Freiburg and are desperate to avoid their fifth straight defeat.