The Bianconeri are interested in adding to their defensive options and have set their sights on the experienced stopper, who has featured infrequently this season
The 35-year-old has only a few months left on the one-year contract which he signed with the Rossoneri in 2011.
It is believed that Juventus are planning to sign the defender if he fails to agree a new deal with Milan in a similar move to the one that took Andrea Pirlo to the club last summer.
Nesta joined Milan from Lazio in 2002 in a €30.5 million deal and immediately became a key member of the team, helping the club win the Champions League in his first season at San Siro.
He also played a pivotal role when the club won the Champions League in 2006-07 and Serie A in 2003-04 and 2010-11.
However, the Italy international has been restricted to only 11 league appearances this season, which may open the door to a new challenge in his career.