The Serbian defender, who joined The Red Devils in 2006 from Spartak Moscow has a wealth of experience eight-and-a-half-year stay and has made more than 200 appearances for the club, helping them to earn five Premier League titles, the Champions League in 2008 and three League Cups.
The 32-year-old centre-back said last month that he would leave Old Trafford when his contract expired at the end of the season.
“Vidic is a world-class player,” Inter president Thohir told the club’s official website.
“I’m extremely happy to have completed this deal to bring Nemanja to Milan. He’s one of the world’s best defenders and his qualities, international pedigree and charisma will be an asset to the team and help bring through our younger players.
“He will bring added value to the club and is another piece of the great Inter side we are building.”
According to reports from Daily Mail, The duo defender Patrice Evra and Rio Ferdinand may leave the club this summer, while Ryan Giggs is expected to retire and the United ace Robin van Persie has been linked with a move back to his previous club Arsenal.