The 32-year-old centre-half confirmed last month that he would leave Old Trafford when his contract expires at the end of the season, and was strongly linked with a move to San Siro.
Inter have now confirmed Vidic's impending arrival, with club president Erick Thohir labelling the Serbian "one of the world's best defenders".
A statement on Inter's official website read: "Nemanja Vidic will wear the black and blue of Inter next season.
"FC Internazionale is delighted to announce that negotiations to bring the Manchester United captain to the club have reached a successful conclusion."
Thohir added: "Vidic is a world-class player. 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.
"I would also like to take this opportunity to thank Manchester United, Nemanja and his entourage for the professional and cooperative approach they have shown throughout the negotiations."
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