Vieira is currently on the staff at Manchester City as an elite development coach but he is on a rolling contract that is up for renewal this summer.
Steve McClaren has been the bookmakers’ favourite to take over from interim boss John Carver, who guided Newcastle to Premier League safety with a final day 2-0 win over West Ham.
But Vieira has been well backed in the last couple of days to become second favourite behind McClaren, who departed Derby at the end of the season after failing to guide them into the Sky Bet Championship play-offs.
According to reports, Vieira is eager to move into senior management and Newcastle are keen to find a young, hungry head coach as they look to rebuild their squad.
Vieira, who played more than 300 times for Arsenal in a glittering playing career, has been completing his UEFA Pro Licence on a course in Wales this week.
Newcastle made a number of player decisions on Friday, with long-serving Jonas Gutierrez and Ryan Taylor among those being released after a season that saw them in a surprise relegation dice.