The Frenchman is out of contract at the end of the season and looks set to end his 10-year affiliation with Manchester United in the summer.
Evra has been linked with moves to Monaco and Paris Saint-Germain in recent months - but reports suggest he is going to reject a move back to his homeland.
The 32-year-old joined Manchester United during the same transfer window as Vidic and looks set to follow the Serbian international to the San Siro.
Manchester United boss Moyes was keen on keeping Evra until the end of the season as back-up to a summer recruit, however the France international wants to experience playing back in Italy.
Evra spent two seasons in France with Marsala and Monza before sealing a move to Nice in 2000.
Moyes is likely to make a move for either Luke Shaw or Fabio Coentrao at the end of the season.