Wenger has reportedly been told he will not continue as manager after the expiration of his contract in the 2018/19 season.
Vieira is an Arsenal legend and part of the 'invincible' team that won the Premier League title with an unbeaten record.
Several candidates have been reportedly shortlisted to succeed the French man when he leaves the Emirates Stadium.
A new report by the has now strongly linked his former player Vieira as Wenger’s successor.
Vieira played for Arsenal under Wenger for nine years between 1996 and 2005 and was named the captain of the team for three seasons.
The report states that even though some Arsenal supporters want Wenger out as soon as possible, the French man may see out his contract.
With a top-four spot already out of reach, the Gunners hope to seal a Champions League spot by winning the Europa League which could determine if Wenger stays or decides to walk away at the end of this season.
The report added that the Arsenal hierarchy wants Vieira as he is familiar with first-team coaches Steve Bould, Gerry Peyton, Neil Banfield and Jens Lehmann.
Still, 41-years-old Vieira is already the head coach of Major League Soccer (MLS) side New York City FC, and previously coached the Manchester City reserve team.
Vieira won three Premier League, FA Cup and FA Community Shield titles during his time at Arsenal.