Inzaghi was forced off during Wednesday's night's 3-1 victory over Palermo that took Milan top of Serie A, just 15 minutes after coming on as a substitute.
But tests on Thursday showed he will need an operation to repair the damage.
It means the 37-year-old could miss the rest of the season, which could also spell the end of his glittering career given that he will turn 38 before the start of next season.
After 289 appearances and 125 goals for Milan since arriving in 2001, Inzaghi has often been a bit part player these last few seasons.
This campaign he has scored an impressive four goals in eight games but mostly as a substitute, actually playing less than three full games.
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