The Frenchman who has been out of work since leaving Arsenal in May has been in talks about taking over at the San Siro, claim France Football.
Wenger recently revealed to Sky Sports he is missing the day-to-day involvement after 22 years at Arsenal and was sure of a return next year.
The 69-year-old said: "I'll be somewhere early next year but I do not know where, I got a good rest. I watched a lot of football, I miss it."
Wenger has been linked with the sporting director gig at Paris Saint-Germain amid reports of unrest between Antero Henrique and head coach Thomas Tuchel.
Under-fire Milan boss Gattuso is hanging on to his job despite a recent upturn in form, winning six of his last eight games.
Billionaire businessman Paul Singer took control of Milan in June and has already recruited ex-Gunners chief executive Gazidis to take up a similar role at the San Siro.
Singer, who has spent a lot time in London watching Premier League football, is a big fan of Wenger and has reportedly given the green light to Gazidis to make contact with the French coach.
And Gazidis, despite his issues with Wenger towards the end of his time in North London, wants his experience to help Milan start to challenge the domestic dominance of Juventus.
Wenger led the Gunners to three Premier League titles, seven FA Cup triumphs and the 2006 Champions League final, where they lost to Barcelona in Paris.
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