City's chances of retaining their crown look slim but Mancini hopes Manchester United's cup exertions can derail their season.
With a 12-point lead and just 10 games remaining, United are firm title favourites but they suffered a blow to their pride in midweek as they crashed out of the Champions League to Real Madrid.
Sir Alex Ferguson's men must pick themselves up for a testing FA Cup quarter-final against Chelsea on Sunday.
Mancini, whose side will be expected to make comfortable progress in the same competition at Championship side Barnsley's expense, said: "It's possible because when United play Chelsea it's possible that Chelsea will beat United.
"It's not impossible in football. One or two weeks can change everything.
"For this reason, we should be confident for the Premier League.
"If they are so strong that they will win all their games then okay, but if they lose one or two games or a point, then we should be ready to take this opportunity."
Although Mancini retains some optimism, the FA Cup does look like City's only realistic hope of silverware.
Some observers may view that as a failure after the dramatic success of last year but Mancini is convinced that would still be a sign of progress given the club's relatively recent lowly stature.
The Italian said: "We started this season wanting to win the Premier League again. We want to try until the last game.
"But if we take second position and win the FA Cup then I think it will be important for us because it's important to win one or two trophies every year."
Real Madrid have now eliminated City and United from the Champions League this season and Mancini feels that gives his side's laboured European campaign a greater context.
He said: "I think we had a very difficult group. Real Madrid and Borussia Dortmund are better than us at the moment.
"It was difficult for us. Maybe we didn't play very well in our group, but it was a really difficult group.
"Madrid and Borussia are in the quarter-finals."
Mancini acknowledges there may be a strong desire around the country for neighbours City and United, the Premier League's top two teams for the past two seasons, to face each other in the FA Cup final at Wembley.
He said: "It's important that we go in the final and if we play against Man United, sure that will be a better game, but also (it would be a good game) against Everton, Wigan or Blackburn.
"For us it's important to go in the final."
City head into their clash against Championship strugglers Barnsley as heavy favourites. Some bookmakers have even quoted odds of 18/1 for a Tykes victory at the Etihad Stadium.
Although Mancini will not be risking key players Vincent Kompany and Sergio Aguero due to injury, he will not be taking the game lightly.
He said: "These games are difficult because they don't have anything to lose and everyone can think it will be easy.
"I have watched Barnsley and it is clear they are not Real Madrid, but these games are strange.
"I don't think many people thought Arsenal could go out in a game against Bradford."
Kompany's absence will take the number of games he has missed since suffering a calf injury to seven despite Mancini saying a number of times he is close to fitness.
He insists there is no major concern over the Belgium defender and it is a case of careful management.
Mancini said: "Calf injuries are a big problem. If you don't recover very well it can happen like it has with 'Vinnie'.
"Maybe he wants to play on Saturday because he always wants to play, but we can't take this risk. We want him to recover well."
Mancini hopes both Kompany and Aguero, who has a knee injury, will be fit for next week's trip to Everton.