"I never had any doubt about it," Vieira explained. I am not 20 but I am not 40 or 45 either. I am still really motivated and believe I can be successful.
"When I look at the players City have got, I think we can achieve a lot. I strongly believe we are not out of the race to win the title but we need to believe the club can do it, because we can.
"I have always been lucky enough to choose the right team to achieve things in the past. I really believe City can do the same and I want to be a part of it.
"The manager knows me really well. I know I can give something to the club and I hope they can give me the opportunity to get back into the France national team."
Mancini has already signed Vieira once, for Inter in 2006, and is confident he can still make an impact in England's top flight.
"He knows me and my staff well, and importantly he also knows what the Premier League is all about," Mancini explained. "He will not need much time to settle in.
"Patrick is one of the great players of his era with almost every honour in the game against his name. I am sure City fans will welcome him warmly and appreciate the contribution he will make."
Vieira, who is still captain of his country, brought his first spell in English football to an end when he left Arsenal for Juventus in the summer of 2005 after nine years at the London club.
Vieira led Arsenal to 2005 FA Cup final glory in his final game for the Gunners
He joined Juventus for £13.7m but spent only one season in Turin before Mancini signed him for Inter, where he won two Serie A titles under the Italian in 2007 and 2008.
Vieira won a third successive Scudetto last term but has made only 16 appearances in all competitions this time round - with most of those coming as a substitute.
Speaking to French television, he confirmed he ended his four-and-a-half-year stay in Italy to help his bid to win a place in the France squad for the 2010 World Cup.
The 6ft 3in central midfielder earned the last of his 107 international caps in a friendly against Nigeria in August but his last competitive appearance came in September 2007 during a European Championship qualifier against Scotland.
"It's over, after a year in Turin and three-and-a-half years in Milan," said Vieira, who helped France win the 1998 World Cup and 2000 European Championship. "I wanted to go to the World Cup and I need playing time.
"I have objectives and I can't achieve them by staying at Inter where I am not part of the coach's choice.
"What's important for me is to go to Manchester City, to show the coach that I am the same as I was during the period when he was at Inter. All this will help me get into the France team."
Archive - Vieira stars for France
Inter boss Jose Mourinho opened the way for Vieira to return to English football after Inter's 1-0 victory over Chievo on Wednesday.
The Frenchman played 90 minutes and gave an impressive display as the Nerazzurri maintained their eight-point lead at the top of Serie A.
"I should give a special mention to Vieira, who certainly played his last game with us," said Mourinho.
"I will miss him but I don't have the right to demand anything from a club who has done so much for me.
"He had an impeccable game [against Chievo]. It was the perfect farewell to us, and I want to wish him all the best for his new career."
Vieira is best remembered for his career in north London, where he made more than 400 appearances for Arsenal, winning three league titles and four FA Cups.
His last game in English football was the 2005 FA Cup final, in which he scored the winning penalty for Arsenal to beat Manchester United.
Gunners boss Arsene Wenger, who was linked with re-signing Vieira at the start of the season, feels he will still be a force in the Premier League and understands his motives for a move.
Wenger said: "For me he has made the right choice and City has made the right choice so it should be a good marriage.
"I thought about it [re-signing Vieira] but I felt that we had players like Alex Song coming through and to keep on giving them a chance it was important that nobody steps in front of them."
Spurs 'were interested in Vieira'
Tottenham boss Harry Redknapp has revealed he almost signed Vieira last summer but chose not to follow up his interest this month.
"I was interested in the summer, very interested and he was keen to come to Tottenham at the time," Redknapp said.
"He showed a lot of bottle to want to play at Tottenham after being such a legend at Arsenal. That shows his character to want to take that challenge on."
"But it didn't happen because he got back into the squad at Inter Milan and it drifted.
"This time when he was available I did get some phone calls about the possibility again but we felt we'd moved on a bit from then and we were happy with what we had at the club."
Vieira's current form has been called into doubt after he was reduced to playing a bit-part role for Inter this season, but Redknapp feels Mancini will not consider it a risk to sign him.
"He's a leader, a great character and was a fantastic player at Arsenal," added Redknapp. "I think any club would want Patrick in their team if he is anywhere near what he was - but only time will tell whether he is.
"But Mancini knows him well. He's no fool and he knows whether he is capable or not."
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