Gomez sidelined for up to three weeks
Gomez suffered the injury in the first half of the 3-2 defeat that saw the Germans reach the quarter-finals on away goals.
"He will be out for two to three weeks," the Bundesliga leaders said in a statement on Wednesday.
Germany international Gomez should be fit for the first leg of the Champions League quarter-finals on March 30 or 31, it added.
Bayern are also missing injured defenders Martin Demichelis and Diego Contento.