The France star was ruled out of the tournament after the problem flared up during his country's training camp in early June.
At the time, France coach Didier Deschamps said that the 31-year-old would be unable to play or train for "several weeks".
Ribery himself came out in June to deny reports that he might need surgery on the injury and Bayern revealed on Tuesday that he is on the road to recovery.
Accompanying a picture of the forward in training, the Bundesliga champions wrote on Twitter: "Welcome back, Franck! #Ribery is completing a fitness course at Sabener Strasse."
Bayern's first competitive fixture of the 2014-15 will come on August 17 as they begin their defence of the DFB-Pokal at third-tier side Preussen Munster.