City have already taken steps in the right direction after their FA Cup win in 2010-11, breaking their run of 35 years without a trophy, soon followed by their dramatic 2011-12 Premier League success.
Vieira, who signed for City manager Roberto Mancini in 2010 before hanging up his boots at the club, believes that they can continue to haul in the trophies like their local rivals United have done in the past 20 years.
"We want to build a dynasty of our own here," Vieira told The Sun.
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"At City we accept that we don't have the history of Manchester United - United will always be United, we know that. We give them every respect for that - they are the biggest club in this league. We can't change that overnight so the only thing we can do is challenge United for the future.
"It will take time - it's not going to happen in one or two years with one or two trophies. But things are changing, we are already getting closer to them. Remember, this season we are the champions. And we want to keep on challenging them.
"We want to fight with them every season to try to win the league, to try to win the Champions League.
"That's why we are carefully putting into place the right project, the right foundations to ensure that we are successful long-term.
"We have made a start. But now we are champions there comes even more pressure. We put more pressure on our own shoulders to stay at the top. There's more pressure from supporters and there's more pressure from outside of the club.
"Every team wants to beat you, there is more coverage, more attention, more interest. You have to be able to handle that, take care of that - even down to the smallest details. So right now we have to concentrate, focus, work hard and then see what happens."