The 32-year-old left United on a free transfer after the club's worst season for over 20 years.
Vidic said: "It is one of the biggest clubs in the world, it has won the Champions Leagues and many domestic trophies.
"It is a new project of the club and new owner, I like the idea of joining Inter and I am sure I made the right choice.
"I spoke to Dejan Stankovic, he was full of support for the club and people around it. Hopefully, I will keep the Serbian tradition at the club. I also spoke to players I know who played in Italy and they said it would be a good move for me.
"I almost came to Italy 12 years ago ... but I don't think it is a step down, I think Italian football is suited to the way I play and I am going to one of the biggest clubs in the world.
"A few clubs are in transition and some are not where they want to be, but they are moving forward.
"Italy is different to English football, but certainly won't be easy in Italy. I was shown the project at Inter and I am happy. They explained the ideas at the club and the way the club is moving, how the club works, and what they are trying to do.
"It seemed very similar to the structure at Manchester United. Erick Thohir wants to build a project and isn't looking at just the next six months. The clubs I have been at before, such as Red Star Belgrade and Spartak Moscow, I have always been part of new projects.
"And In my career I have always chosen clubs which are moving forward, and I hope to achieve at Inter what I achieved at United."
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