European player of the year Ribery was on the receiving end of a series of crunching tackles as France beat Ukraine 3-0 on Tuesday for a 3-2 aggregate victory and a World Cup next year.
"He suffered a crack in his 10th rib and will definitely not play against Borussia Dortmund on Saturday," the club said in a statement on Thursday.
"The 30-year-old's participation in the Champions League group match at CSKA Moscow next week is also in doubt."
Bayern, unbeaten in 37 league games, top the Bundesliga on 32 points, four ahead of second-placed Dortmund.
"This is bitter as I would have loved to be there for this top game. But I am certain the team will be successful without me," Ribery said.
While his potential absence in Moscow is of no major significance given the Bavarians have already qualified for the knockout stage with two games to spare, he will be sorely missed in Dortmund. Bayern are already without playmaker Bastian Schweinsteiger.
Bayern, who beat Dortmund in last season's Champions League final, are still waiting for their first league win against their title rivals after six failed attempts since 2010.
Dortmund have their own injury woes, having run out of first choice defenders following injuries to Mats Hummels and Marcel Schmelzer in Germany's 1-0 win over England on Tuesday.
(Reporting by Karolos Grohmann; editing by Justin Palmer)
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