The Gunners chief executive has confirmed that he will be joining Milan on December 1, having been offered a lucrative contract by the Italian outfit's new owners.
Gazidis was largely credited with restructuring Arsenal's backroom staff in recent years, in addition to being a driving force behind the exit of Arsene Wenger and subsequent arrival of Unai Emery at the club.
Football.london reports that one of his first tasks at Milan will now be to coax Ramsey to San Siro, with the Wales international still embroiled in contract talks at the Emirates Stadium.
Ramsey's current deal expires at the end of this season, meaning that he could leave on a free transfer next summer or for a cut-price deal in January.
A number of clubs - including the likes of Manchester United and Chelsea - are understood to be monitoring Ramsey's situation in North London and could provide competition for Milan, who would be free to open talks in January if no new deal is agreed with Arsenal before then.