France international Vieira, who becomes new Blues boss Roberto Mancini's first signing, has signed a six-month deal at Eastlands with an option for a further year.
"When you look at the players the club has got you can achieve a lot," said the France international. "I strongly believe we are not out of the race to win the title. I am sure we can do it."
The 33-year-old former Arsenal midfielder, whose deal at Inter was due to expire in the summer, also insisted City were not getting a player past his best.
"I'm really delighted, really happy, really proud to be back. City are a really ambitious club," he continued. I am very confident, I believe in myself.
"I know I am not 20, but I am not 40-45. Overall I am really motivated. I know I can give something to this club and hopefully we will be successful this year."
Vieira admitted the move was motivated in part by his desire to secure first-team football to work his way into the France squad ahead of the World Cup finals.
"I hope that the club as well will give me the opportunity to get back to the French national team," he added.
Vieira feels City are a club which are capable of winning trophies. He added: "I chose City because I really believe something can be achieved at this club and I want to be part of it."
Mancini, who coached Vieira at Inter, said he saw the midfielder as a regular player. "Patrick for me is a big player, an important player," said the Italian.