Relegation-threatened Sassuolo are searching for a new coach after parting company with Eusebio Di Francesco on Tuesday following a 3-1 defeat to fellow strugglers Livorno on Sunday.
And Galliani revealed that Milan rebuffed an advance from Sassuolo to hire Milan youth-team coach Inzaghi - who had been linked with the role of first-team coach at San Siro prior to the appointment of Clarence Seedorf - as Di Francesco's replacement.
"Sassuolo have asked us (about Inzaghi), and we said no," Galliani said. "My dear friend Pippo came to our headquarters this morning and we told him we cannot let him go.
"I'm sorry for Sassuolo, whose president is a friend of ours and a Milan supporter, but Pippo is learning his trade here as coach of our youth team.
"He is a Milan legend and so he's not going away from here. He's got a contract until 2016. But I can confirm that Sassuolo asked us for him."
Sassuolo sit 18th in Serie A with just 17 points from 21 games ahead of the visit of Verona on Sunday.
Inzaghi accepted a coaching position at Milan in 2012 following 11 years with the club as a player, a spell in which he won two Serie A titles and the UEFA Champions League twice.
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