Vidic announced at the beginning of last month that he would leave Old Trafford when his contract expires this summer, and Goal later reported he had agreed to join Inter.
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"Vidic is a world-class player," 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.
"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."
The signing of Vidic is the first step in Inter re-modelling their defence, with United team-mate Patrice Evra and Arsenal full-back Bacary Sagna also summer targets.