The 34-year-old hurt his left knee in the 2-2 draw at Hertha Berlin on Sunday and scans have shown he has suffered a lateral knee ligament (LCL) tear.
The former France international will now not be available for the Group B visit of Celtic to the Allianz Arena on 18 October and the return clash in Glasgow at the end of the month.
The managerless Bundesliga club tweeted: “Breaking: @FranckRibery has suffered an LCL tear in his left knee. Get well soon, Franck! #ComeBackStronger #MiaSanMia”
Bayern have not put a timescale on Ribery’s potential absence, other than to say he will be sidelined for the “coming weeks”.
He will not undergo surgery at the moment, with the club instead treating the injury conservatively. Chief executive Karl-Heinz Rummenigge said in a statement on the club website: “We’re sorry Franck has hurt himself. We all wish him a quick and good improvement.”