Gazidis is the Rossoneri's favoured choice to replace general manager Marco Fassone, who was sacked in July. Milan want the 53-year-old to help develop their brand and deal with UEFA financial fair play regulations.
He has been offered a big salary and is close to reaching an agreement with the Serie A side, according to La Gazzetta dello Sport (h/t Calciomercato.com).
Gazidis was named Arsenal's chief executive in November 2008 and took up the role in January 2009. He had worked previously as the deputy commissioner for Major League Soccer.
He has been linked with a move to AC Milan throughout the summer. Both the club and manager Unai Emery have become frustrated with the uncertainty over his position, per Matt Hughes at The Times.
Emery said earlier in the summer he expected Gazidis to stay, per Goal's Chris Wheatley:
The Gunners released a statement in response to the speculation surrounding Gazidis saying he was "fully committed to taking Arsenal forward."
Yet the speculation persists, and Gazidis could leave. He earns £2.6 million annually at Arsenal, but AC Milan have offered him an extra £1 million a year, according to Matt Law at the Daily Telegraph.
The Rossoneri were taken over by American hedge fund Elliott Management in July, which pledged to inject €50 million in capital into the club, per Football Italia.
AC Milan invested heavily in the squad over the summer on players including Mattia Caldara, Samu Castillejo, Gonzalo Higuain and Diego Laxalt.
The club are keen to return to the top of Italian football after several disappointing seasons. Their last title came in 2013, and they have finished outside the top four for the last five seasons.
Gazidis could play an important role after becoming increasingly influential at Arsenal. He oversaw Arsene Wenger's departure and was involved in bringing in Emery.
It's a new era at Arsenal after Wenger, but they face a tough battle to keep up with the likes of Manchester City, United, Chelsea and Liverpool who all finished above the Gunners last season and have not been afraid to spend heavily over recent seasons.
A move to Italy could well tempt Gazidis and not solely for financial reasons. There will be more scrutiny on Serie A this season, as Portuguese superstar Cristiano Ronaldo has joined Juventus.
AC Milan and Gazidis could look to take advantage of a potential increase in popularity and interest in the league, as they bid to restore one of the biggest clubs in Europe to its former glory.