The Frenchman, 47, could also reunite with Arsene Wenger in Italy after reports claimed the Gunners legend is line to replace Gennaro Gattuso.
Grimandi has been working for the Gunners since 2006 and he spent five years with the club as a player.
But according to Daily Cannon, the scout is set to continue his career with the Serie A giants.
Gazidis is set to start work with AC Milan later this month after he left his role as Arsenal's chief executive after ten years.
The 54-year-old wants to take Grimandi with him to the San Siro following their formidable partnership in North London.
Gazidis played a key role in hiring Unai Emery as Arsene Wenger's replacement, but he wants to work with the Frenchman again.
Reports suggest that the new AC Milan chief wants to hire Wenger as the club's new manager.
The Gunners legend has been out of work since leaving the club in May and has admitted that he is raring to go.
Wenger confirmed in October that he will return to football in the New Year, but ruled out of the option of a Premier League job.
Grimandi spent most of his time at Arsenal working with Wenger and reports suggest he would jump at the chance to work with the 69-year-old again.
The scout started his career as a player with Gap before he joined Monaco and then Arsenal in 1997.
He scored five goals in 155 competitive appearances for the Gunners.
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