That is according to Italian newspaper Corriere dello Sport.
They claim Gazidis could use his Arsenal connections to make a deal at the San Siro happen.
Gazidis is set to take up a new role as CEO at Milan next year.
The director oversaw the departure of Arsene Wenger as Arsenal manager in the summer, and the subsequent appointment of Unai Emery.
Aaron Ramsey is out of contract at Arsenal next summer (Image: GETTY)
Aaron Ramsey can speak to other clubs in January regarding a free transfer (Image: GETTY)
Ivan Gazidis will join AC Milan in the New Year (Image: GETTY)
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Corriere dello Sport claim he has already been in touch with Milan director Leonardo to discuss the new project at Milan.
The current international break is being used to assess manager Gennaro Gattuso’s running of the team and where it needs strengthening.
The report claims Milan are also interested in PSG midfielder Adrien Rabiot.
His contract also expires at the end of the season and could discuss a free transfer come the New Year.