The Premier League champions are in the middle of a slump, after a 3-1 defeat at the hands of struggling Southampton on Saturday at St Mary's.
The loss has left City 12 points behind rivals Manchester United in the title race, and the Italian has lost the dressing room after criticising his players following the game, according to the Daily Star.
Vieira, who currently works as the football development executive at the Etihad Stadium, played under Mancini at both Inter Milan and City but has yet to gain any coaching experience.
The Arsenal legend was linked with a return to the Gunners as the head of youth development recently, but it looks as if he could make a swift move to management.
Mancini took over at the club in 2009.