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The Italian side recently got taken over by US hedge fund Elliott Management, who have promised to inject €50m (£44.2m) to help stabilise the club’s finances.
They are also looking to appoint a new president and board of directors as they continue to revamp Milan.
And, according to Sky Sport Italia, Arsenal chief executive Gazidis is one of those the US hedge fund want to bring in.
The 53-year-old has been with the Gunners since 2009 after previously working as the MLS’ deputy commissioner.
Gazidis helped bring new head coach Unai Emery to Arsenal this summer and now has a key role in the running of the club.
However, Elliott Management are hoping he would make the jump to Italy to help rebuild Milan.