City won a first Premier League title in 44 years in 2012, but have yet to make a firm impact on the Champions League, in which they face the might of Barcelona later this month in the first knockout round.
Mourinho was manager early in the Roman Abramovich revolution, guiding Chelsea to a first championship in 50 years, something they successfully defended, while also winning the League Cup twice and the FA Cup in a period bankrolled by the billionaire Russian oligarch.
The early trophy return following Abramovich's outlandish spending compares favourably with Sheikh Mansour's outlay under Roberto Mancini, who is now in charge of a Galatasaray team Chelsea face in Europe this month.
"They won one title, won a couple of cups," said Mourinho, the only Champions League-winning manager in the Premier League.
"Only in Europe they didn't do well, or close to doing well.
"Speaking objectively, they did very bad in the Champions League in previous seasons, also in the Europa League.
"But the team is fantastic, the squad is fantastic and normally they (should) win more titles."
Asked what it will take for City to progress to the latter stages of the Champions League, Mourinho made reference to boss Manuel Pellegrini's mis-calculation at Bayern Munich, when his side needed one more goal to advance as group winners and avoid the likes of Barcelona in the last 16.
"The first thing to be successful in Europe is to know the rules of the competition. That's the first thing," added Mourinho, with a smile.
Compared to his first spell, Mourinho is working in straitened times, with Chelsea insisting they are complying with UEFA's Financial Fair Play rules.
Mourinho has repeatedly taken veiled swipes at City for, in his eyes, flouting the regulations which are still to be enforced with sanctions, and is enjoying the challenge of building a Chelsea team for the next decade.
He believes City are clear favourites for the Premier League title and must accept that responsibility, but that Chelsea will soon be ready to go toe-to-toe with them.
He added: "I enjoy building a team. I enjoy building the future.
"I enjoy not working just for today, and next season to start again with another team, spending a lot of money again, selling a lot of players, bringing in a lot of players.
"I'm enjoying this very, very much. To be the underdog I don't enjoy.
"(But) I think I'm going to enjoy next season. It is more me - and I think it's more Chelsea too - to start next season and say immediately we are candidates, the same way the others are candidates."
Asked if Chelsea can keep pace with City, given their spending, Mourinho said: "We think that we are not very far from having a squad that allows us to compete face to face with every club, doesn't matter how much the investment is, doesn't matter how much the financial play is 'fair' or 'dodgy'."
Mourinho is guiding Chelsea through a transformation which will see the likes of Frank Lampard, John Terry and Ashley Cole end their Blues careers.
The Blues boss has repeatedly insisted the trio are relaxed about their contracts expiring at the end of this season and are in no rush to begin negotiations over a new one.
Cole has had limited playing time of late due to the form of Spain right-back Cesar Azpilicueta at left-back.
Mourinho does not believe that will impact on Cole's status as England left-back.
"Ashley is playing well with us," Mourinho said.
"If he's not playing every game, okay, he's not playing every game, but he's playing and he's playing well and I think that's what Roy Hodgson needs to feel and know.
"I believe more in stability, believe more in knowing the player and what the player can give you in high-pressure situations like a World Cup.
"It's not my problem, it's Roy's problem, but I think much more that he goes for stability and takes Ashley and (Leighton) Baines that were the two left-backs in qualification."