United had been the dominant force of English football throughout the Premier League era until Sir Alex Ferguson's retirement in 2013, since when they have failed to finish in the top three.
City once again lead their local rivals this season heading into Sunday's derby at Old Trafford, but Kompany is wary that United's blip could only be temporary as they come to terms with life after their most successful manager.
"I have been a part of trying to turn this club around. When I signed for City we were out and about in town a lot because we weren't the big dogs in the city. That's changed a bit now. We're now more like an international club, but it wasn't like that when I signed. When the owners came to Manchester City they decided they wanted to make this club a huge club in England. At the same time, there was a huge gap left when Ferguson left United," he told The Guardian.
"It's been a case that as one Manchester club was going through a rebuilding phase, we were in the ascendancy. But one thing I would like to say, touching on Ferguson, is that it is absolutely normal that when a person has had such a big influence on a club for such a long time, there will always be a big rebuilding job to do once he has left.
"That's why I am reluctant to say there has been a shift in power. I'm not defending United. They have made a huge investment to try to close the gap on us and fill the void left by Ferguson. But it is normal that it would take them a few years to steady the ship once he left."
City are bidding for a record-equalling 14th consecutive Premier League win on Sunday, and doing so would lift them 11 points clear of United at the top of the table.