United are second and 16 points behind champions-elect City in England, but Jose Mourinho's men are in the last 16 of the Champions League.
And Vidic — who won the Champions League and five Premier League trophies during his eight years in Manchester — feels United are equipped to succeed in Europe.
"It's Man United. I know the Premier League is bigger, but I think the Champions League could happen," Vidic said via ESPN.
"When the season started, I said that I thought United had more chance to win the Champions League than the Premier League because the players they signed didn't have the time to fit in.
"Also, Mourinho is a tactical coach, knows Europe very well. He knows there are going to be scrappy games and how to get through them.
"That's my belief. To win, you need a bit of luck, but there is a chance to get to the semi-final or final. You never know. And if you win a Champions League, the players start to believe. They think: 'OK, speak about me now.' Phil Jones, Chris Smalling, they will look different than now."
Vidic, who said he wants to become a manager, was speaking ahead of United's showdown with rivals Liverpool in the Premier League Saturday.
United are second and two points clear of Liverpool, and Mourinho's side can extend that gap with victory at the Theatre of Dreams.
Previewing the mouth-watering clash, former Serbia international and United favourite Vidic again spoke about how he almost joined Liverpool before moving to Manchester in 2006.
"In Rafael Benitez's time. I spoke to him about a few things, but my wife was the main one who spoke to him," the 36-year-old added. "He was interested in signing me, but I don't know what was the problem at the time. I spoke to him before I signed for Manchester United, though. I didn't meet, but we spoke over the phone, in my car in Moscow, with my wife translating.
"I didn't say no. They said just that they were interested, they want me. It was going to happen, going to happen. Then someone came and said: 'Do you want to play for United?' and I said yes. After two days, everything was done. Then Liverpool said they wanted to do the deal, but everything was done."