The Uruguayan centre-back has been offered a contract extension by the La Liga side, however he is yet to sign the deal offered to him. With his current contract set to expire at the end of the season, the 32-year-old could leave Atletico for free in the summer.
According to the Corriere dello Sport, Rossoneri technical director Leonardo is already trying to secure Godin's signature for next season, in a move that would bring a wealth of experience to the Milan backline.
The 32-year-old has been a vital part of the Atleti defence, making over 250 appearances for the club since moving to Madrid from Villareal back in 2010. He turned down a move to Manchester United last summer, as he wanted to try and reach a Champions League final at the Wanda Metropolitano this season.
Godin, who has 125 caps for Uruguay, has played 10 times for Atleti this season, scoring twice, including a last-minute winner for the Madrid side in their dramatic 3-2 win over Athletic Bilbao in their last La Liga fixture.
The win moved the Rojiblancos up to third place in the league table, just one point behind leaders Barcelona, who Atleti play at the Wanda Metropolitano on Saturday in a huge opportunity for Diego Simeone's men to go to the top of the table.
The Rossoneri's four-game unbeaten run in the league ended with a 2-0 defeat to Juventus at the San Siro before the international break. Milan currently sit fifth in the Serie A, as they try to make up ground after a poor start to their league campaign.
This weekend Milan face a tricky trip to fourth-placed Lazio, with Gennaro Gattuso's team given an opportunity to leapfrog their rivals in a direct clash.