Wenger is impressed by what he has seen from a man who has moved “intelligently and patiently” since hanging up his boots and believes he will one day fill a role that he once did in north London.
Vieira is delighted with that recommendation to Arsenal, telling Press Association Sport: "I'm really privileged to have worked under Arsene and him saying that is really flattering.
"If I manage Arsenal one day I will take him for dinner and I will pay the bill!
"He didn't say it to put pressure [on me]. It's just flattering that the guy who spent over 20 years there thinks I have the quality to manage a big club like Arsenal one day."
Vieira added on his future ambition, with there a desire on his part to one day work in the Premier League or Serie A: "Since I was in Manchester City I had a clear idea about the next step for me.
"Moving from City to New York in the MLS and then coming back to the French league was the right decision.
"I have a clear idea of what I want to achieve, and I will try to take the right decision to get to the top.
"I want to take my time to get there, but I would love one day to manage one of the teams I played for in Italy or England.
"When you have an exciting league like in England, you always want to get involved. But I don't put a time on it, because my work will decide if I deserve it or not.
"That is what will dictate if I deserve this kind of job."
For now, Vieira is happy to continue on a learning curve that he hopes will lead him to the top.
"There is more pressure in Europe than in America but I really enjoyed all the difficulties," said the 42-year-old, who has just completed a first full season back in European football.
"I'm finding my way but we [Nice] finished the season seventh with the third youngest team in the league, with all our injury troubles on the field and no number nine.
"It's been challenging, but we did a good season. The progress is never enough because we always want to do better, but to finish in that position is really good."