The Serbia international, who announced in January that he intended to leave Old Trafford this summer, will sign a two-year contract worth 3 million euros (potentially rising to 3.5 million euros) per year.
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Piero Ausilio, Inter's general manager, has been in close contact with Vidic's people since January and spoke in glowing terms of the defender after he made his announcement to leave United.
Ausilio was in attendance for the Premier League clash against the Gunners, after which a gentleman's agreement was struck between the two parties.
Erick Thohir, the Inter chairman, views Vidic as a potential ambassador for the club and is looking to use the player's vast experience as a means to engage his home nation's interest in soccer and open up the Indonesian market.
Vidic will become the Serie A side's first signing of the summer and is seen as a sensible piece of business, adding experience, quality and value for money given no transfer fee will be paid.
Monaco and Galatasaray showed interest in Vidic but the center back judged Inter's offer to be the most appealing.
Vidic moved to Old Trafford from Spartak Moscow in 2006 and has won five Premier League titles and the Champions League in his eight years with the Red Devils.