Milan owners Elliott Management want to bring Gazidis to Italy to take charge of the club.
Il Giornale says a contract offer has been made worth a clear €1.2m-a-year more than what Gazidis is on at Arsenal. The deal is worth €4m-a-year.
Gunners owner Stan Kroenke is eager not to lose Gazidis, who has overseen massive changes to the club’s management structure over the past 12 months.
Milan have asked Gazidis to respond to their offer over the next month.