City begin their second successive season in the continent's elite competition as the English champions having continued their remarkable recent progress by winning the Barclays Premier League last May.
That came after FA Cup success in 2011 brought the club a first trophy in 35 years, but while Europe might seem the next logical target, Mancini is setting no bold targets for this particular season.
Speaking at the Bernabeu Stadium ahead of his side's mouth-watering Group D opener against Real Madrid, Mancini said: "If you drive a Ferrari you can win, if you drive a Fiat Cinquecento, probably not.
"In the past we have been working for this. I think we beat good teams but we have played together for two years and we needed time.
"I am sure we will arrive to win this trophy in the future. I do not know when, but I am sure because we are working for this.
"If we want to improve every year it is possible."
City failed to progress beyond the group stages last season having been given a tough draw against Bayern Munich, Napoli and Villarreal.
This year the assignment is even tougher with the champions of Holland and Germany, Ajax and Borussia Dortmund respectively, also lying in wait as well as nine-time European champions Real.
Mancini said: "I think it was a good experience last year - we did a good group but it was not enough.
"I think the group will be stronger this year but it will be difficult for all the teams.
"I think it is important now that we go to the second stage. After February anything can happen, but now it is difficult to say what we can do."
Much has been made of Mancini's rivalry with his Real Madrid counterpart Jose Mourinho, but the Italian insists it is not a factor.
The pair have actually only come up against each other in one previous European tie, when Mancini's Lazio met Mourinho's Porto nine years ago.
Despite that, it is assumed tension has existed between them since 2008 over the manner Mourinho succeeded Mancini as Inter Milan boss.
Mourinho then went on to take Mancini's Serie A winners to Champions League glory.
But the Italian said: "We don't have a personal relationship. I met him two times for Porto against Lazio and I met him in Milan but I have only met him four or five times.
"I know him as a manager.
"I think tomorrow [Tuesday] will be a good game, Real Madrid against Manchester City, but I think the most important thing will be the fantastic players on the pitch."
City are yet to hit their stride domestically having been held to draws by Liverpool and Stoke and laboured to wins over Southampton and QPR.
But their performances better those of Real Madrid, who have won just one from four and were beaten at Sevilla at the weekend.
Mancini said: "I watched their game - it was a difficult game - but when you start a season after a European Championship or a World Cup, the first six, seven, or eight games are difficult - for us, Real Madrid, for every team.
"I think we can have a chance. We should be focused on the game and our play."
Striker Sergio Aguero, who was linked with a switch to the Bernabeu last season, appears to be winning his fitness race to play.
The Argentinian has been out for a month with a knee injury suffered in the Premier League opener against Southampton.
Mancini said: "Sergio is okay he is not 100% because he didn't play any games but his feeling at this moment is good and he has trained with the team for the last five days. He has a chance."
City also go into the game lifted by the news that Spain playmaker David Silva has signed a new five-year contract at the Etihad Stadium.
The club were anxious to tie the 26-year-old to a new deal after his outstanding role in last season's title win.
Silva, signed from Valencia in 2010, said: "I'm very pleased because I've been here for two years now and I feel very happy here at the club, in the city and with all the people.
"We've won the Premier League title and two other trophies too, and the team is growing and we are now aiming at the Champions League, and so that also is another reason to want to stay here and try and win it."