The Premier League champions were eliminated by Barcelona in the last 16 of the Champions League last season and have yet to make their mark in continental competition since becoming a major force domestically.
But Kompany, who signed a new five-year deal on Wednesday, believes it is time for City and other Premier League superpowers to pose more of a challenge to European heavyweights Bayern Munich, Real Madrid and Barcelona.
"English clubs need to push again in the Champions League," he is quoted as saying. "I’m quite clear and honest about it that there are three clubs, no more, just three - Bayern Munich, Real Madrid and Barcelona - who are a step ahead of everyone.
"But clubs like City, Chelsea... all the contenders are mainly in England, pushing to change that order.
"It was the case in the Premier League as well. It was Liverpool, United, Arsenal and Chelsea. But now it has been changed. It will happen in Europe.
"We are hoping to change the order in Europe. It’s our ambition. I don't think anyone at City has hidden from it.
"People will always throw the past and the glory back in our face but then again I would like nothing more than to upset the order. Ultimately, our fans have been there a long time and they have a right, just as much as any fans, to be part of that top tier.
"People complain when they get moved off their throne. I find it really funny. You complain, you set up new rules, but ultimately what you are doing is just trying to make sure that the established order doesn't change."
City get their Premier League campaign under way against Newcastle United on Sunday.