Vieira, who turns 35 next month, signed for City 18 months ago as a back-up option in midfield, but he managed 34 appearances for Roberto Mancini's side this season.
"I really want to stay here because I've loved my time with City and the fans have been fantastic. I just need to speak to the manager and then with the board and try to find a solution that suits everyone, but if there is a possibility, I would love to stay here." he said on the club's website.
"It's really nice because although I don't play that much, I've felt a real connection with the fans, particularly over the last month,
"They've really made me feel happy and at Bolton last week it was special because they were singing to me while I warmed up and I really enjoyed it, so I'd like to thank them.
"We are starting to talk about the future only now because the target we had was to qualify for the Champions League and win the FA Cup first, now we have achieved that so I will sit down and discuss things with the club and a decision will be made very soon."
Source: PA
Source: PA