Ramsey will be out of contract at the Emirates next summer, and has yet to agree a new deal.
Widespread reports in Italy now claim the Serie A giants plan to swoop for the Wales international.
And Milan are hopeful of recruiting Arsenal chief executive Gazidis as they reshape their off-field structure.
Gazzetta dello Sport says the deal is close and it is understood Milan are prepared to pay Gazidis what he wants.
The Italians believe the South African's presence at the San Siro will boost their hopes of landing Ramsey.