The emergence of the Blues, once dubbed “noisy neighbours” by Sir Alex Ferguson, has seen them overtake the Red Devils in the Premier League pecking order.
A record-breaking title win was taken in last season – a second since United last took the crown and a third in total – and Pep Guardiola has his team back in the hunt this term.
Liverpool are also very much in that battle, as they seek to secure domestic dominance for the first time in 29 years.
They are due at Old Trafford on Sunday, having first faced Bayern Munich in the Champions League, and Scholes says that fixture will mean more to United than dates with City.
The Red Devils legend told Sky Sports: “I think it's the biggest rivalry in English football.
“In a few years, if Man City carry on the way they are going, that [rivalry] might overtake it a little bit.”
United took their title haul to 20 under Ferguson, with Liverpool overtaken on that roll of honour as they remain stuck on 18.
Scholes believes that ensuring the Reds remain second on that list continues to give added weight to meetings between two fierce North West rivals.
He added: “You'd expect the two Manchester teams to have the biggest rivalry, but I don't think it is.
“I think it's the Liverpool-Man United thing, especially with Liverpool trying to get to United's record of league titles. We don't want them to do that.
“United vs Liverpool is a big, big game and Liverpool are going really well; so are United. Hopefully they'll beat Liverpool on Sunday.”
United suffered a 3-1 defeat to Liverpool at Anfield earlier in the season, with that result spelling the end for Jose Mourinho as he was relieved of his managerial duties.
Ole Gunnar Solskjaer was appointed as his successor in an interim role and has taken in 11 wins, one draw and a solitary defeat across his 13 games at the helm.
The most recent of those outings came against Chelsea in the FA Cup, with the Red Devils heading into a clash with Liverpool buoyed by a 2-0 victory at Stamford Bridge and a place in the quarter-finals.