On Saturday, the teams meet at Eastlands, with big-spending City looking for their second win of the Premier League season after amassing only four points from three matches so far.
Mancini's biggest challenge remains gelling his expensive recruits into a team capable of winning trophies.
New Blackburn signing Benjani Mwaruwari, who was released by City this summer, was critical of his former club last week, claiming there is no trust and that some of the players are motivated primarily by money.
Allardyce, meanwhile, used the examples of Kenny Dalglish guiding Blackburn to the Premier League title in 1995 and Chelsea's success under Jose Mourinho as managers who have used financial clout to bring great rewards.
The Rovers manager said: "Those are his (Benjani's) words and he's been in the middle of it. If that is the case then as a manager you've got to take control of that and make sure it reverses into a thriving football club of great quality.
"If he (Mancini) gets it right, as Jose got it right for Chelsea after the huge amount of spending that Roman (Abramovich) did, and if you look back here with Kenny, he got it right and Blackburn won the Premier League.
"You can buy a league now, obviously Blackburn have proven that, Chelsea have proven that, and you can do it in a short period of time. If you manage the players right and get that team spirit right, then you'll win anything."
Rovers defender Michel Salgado, meanwhile, believes City fans must be patient and not expect instant success.
The Spaniard spent 10 seasons at Real Madrid, another club famed for big spending, but does not believe City can be compared with the Primera Division giants.
"The difference between Manchester City and Real Madrid is Manchester City so far is a fashion club," said the 34-year-old. "They are building a big club but Real Madrid is a historical club with big pressure to win a title every year.
"They have to spend money to battle with the best teams in Europe. Maybe Manchester City in the future can get closer to Real Madrid but now it's a different story.
"They are signing eight new players every year and that's a problem. First of all you have to build a team.
"The most important thing is for the manager to try to control the changing room. To get everybody happy is very difficult when you are dealing with big players."
Blackburn go into the clash looking to avoid a third consecutive Premier League defeat for the first time since Allardyce succeeded Paul Ince in December 2008.
After opening their campaign with victory over Everton, Blackburn have since lost to Birmingham and Arsenal, and the Rovers boss knows they will have a tough battle on their hands to break that sequence on Saturday.
Allardyce said: "We're absolutely delighted with the performances we have given, unfortunately the performances haven't given us the amount of points we should have got.
"We should have have six or even seven but we haven't done that so we've put ourselves under a huge amount of pressure now by going into the Man City game on the back of two defeats.
"I was told in my time here we've never lost three Premier League games on the trot, which I found quite remarkable. I don't want to lose that record.
"At Eastlands it's always very, very difficult. We got well stuffed 4-1 there last year and they've added another £200million to that squad."
The match will be Blackburn's first since the transfer window shut and Allardyce is still ruing their failure to recruit Bulgarian striker Ivelin Popov.
But Rovers' forward ranks have been boosted not just by the arrival of Benjani but also the return to the squad of Jason Roberts, who has resolved his differences with Allardyce after failing to secure a move away.
They could both play but Keith Andrews (groin) is out and Martin Olsson (shoulder) doubtful.