Gladbach, currently in fourth place and aiming for a surprise Champions League spot, confirmed they would be without the striker for about three weeks.
The 24-year-old Germany international, who joined from Augsburg this season, is to wear a protective bandage for about 10 days to help speed up the healing process.
"He will not only miss Saturday's friendly against Kaiserslautern but most likely the next three league matches," Gladbach said.
Gladbach are away to Stuttgart on Jan. 31 when the Bundesliga returns from the winter break.
Hahn, capped once by Germany, has scored six goals in 23 games for Gladbach in all competitions this season and set up another five.
(Reporting by Karolos Grohmann; Editing by John O'Brien)
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