Galli was a goalkeeper for the Rossoneri and is currently working for Fiorentina, another of his former clubs.
"We haven't made a decision yet. Directors can't change clubs during the same season, so we will make a decision at the end of this campaign," CEO Adriano Galliani told Sky Sport Italia.
"Galli is a Milanista, a good man and has excellent scouting credentials. We will certainly be bringing in a director of sport."