The Serbia international will join Serie A outfit Inter when his current contract expires at the end of next month, ending an eight-year association with the Red Devils, and leaves off the back of the club’s worst season in the Premier League era.
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"We argued amongst ourselves," Vidic told the Daily Telegraph. "This year more than any other, because when you have bad times, people show they care.
"We are still friends, but we were arguing to get better. We wanted to improve.
"We could say those things to each other because we have been together for so long, but it hurt.
"If you didn’t argue, it would not be right. We had some hard moments in the dressing room between ourselves."
Vidic, who made just 23 Premier League starts in his final campaign at United, felt the players were uncomfortable with Moyes’ style and were relieved to get back to the ideas extolled under Sir Alex Ferguson, which returned when Ryan Giggs was placed in temporary charge
"There was a transition," Vidic continued. "You get someone who sees football in a different way and he will want to put his stamp on the team and the way he wants to play.
"Ryan [Giggs] shares the same ideas as Sir Alex Ferguson and his was a more similar approach to the one we had with Sir Alex. The players are more used to it and felt more comfortable with it.
"I am not saying that the David Moyes way was bad, but these players feel more comfortable playing a certain way of football. You have to respect where you are and what you represent, though, and there is no point speaking about someone who was here, who everyone knows lost his job because he did not succeed in doing what he wanted to achieve.
"The best answer I can say is that he [Moyes] tried really hard, he was professional. He was really committed to the job and desperately wanted to do well. But unfortunately, it didn’t happen and we are all sad.
"Sometimes you have bad times and you question yourself, of course, but we never did anything to question the manager at all. We questioned ourselves more."