The Bavarian powerhouse will be without the France winger in Wednesday's match as the club aims to book its place in the quarterfinals.
The France international featured for the full 90 minutes in Bayern's 3-2 Bundesliga win over Fortuna Dusseldorf at the weekend and even scored the equalizer to make it 2-2 in the 73rd minute.
Nevertheless, he complained about pain in his left ankle after the match, and Bayern has now confirmed that he will not be match fit in time for Wednesday's clash with the Gunners.
Ribery's injury is yet another blow for Heynckes, who will also be without Bastian Schweinsteiger and Jerome Boateng due to suspension.
Bayern is defending a 3-1 lead from the first leg at the Emirates Stadium.