Thauvin, 27, is out of contract next June and can agree terms with a new club in January.
The Rossoneri met with his agent Cyril Rool at the beginning of October and remain in contact.
Talks have progress well with Thauvin viewed as a potential replacement for Samu Castillejo, who could return to Spain.
Although Thauvin could be signed for free in June, Milan have not ruled out an offer to take him in January.
Thauvin has made an strong start for OM in the new Ligue 1 season with three goals and four assists in seven games.
He both scored and assisted a goal in Marseille’s 3-1 win over Bordeaux on Saturday.
French international Thauvin made only two appearances last season due to injuries.