The 28-year-old, who made 46 appearances in all competitions last season for the German champions, was injured during training on Saturday.
Subsequent scans revealed a torn ligament and Rafinha is now facing a spell on the sidelines.
"I'm obviously very disappointed," Rafinha told Bayern's official website.
"I had a good preparation, and now this. I am very, very sad."
Bayern play their first competitive match of the new campaign on Wednesday when they face Borussia Dortmund in the DFL Supercup.
Pep Guardiola's men will then take on Preussen Munster in the DFB-Pokal before beginning the defence of their league title at home to Wolfsburg on August 22.