"After a consultation the doctor has confirmed the need for surgery," a Milan statement said on Thursday.
Inzaghi, whose contract expires at the end of the season, suffered the injuries after coming off the bench in Wednesday's 3-1 win over Palermo which sent the Rossoneri top of Serie A.
Milan decided against offering Brazil's Ronaldo a new deal when he suffered a similar injury two years ago but the club's esteem for Inzaghi suggests an extension may be available.
"It can't end and it will not end. Pippo is in our hearts and we in his," Milan's website said.
"I don't give up. I will do everything, believe me, because it is not over. I will also do it thanks to your strength and your energy," he said in a statement to fans.
He came off the bench to score twice in the 2-2 Champions League draw with Real Madrid last week and had been hoping for a role in Sunday's Serie A game with champions Inter Milan.
With Milan top and aiming for their first scudetto since 2004, owner and beleaguered Italian prime minister Silvio Berlusconi may seek a replacement for Inzaghi in the January window to boost his side's hopes and his own popularity.
