The-43-year-old won three Premier League titles and two FA Cups during his nine years with the North Londoners before leaving for Juventus in 2005.
Vieira was captain of the Gunners side that went unbeaten in the 2003/2004 Premier League season, which was the last time they were champions of England.
Following a maiden managerial spell with New York City in USA, the Frenchman is currently in his second season with Ligue 1 outfit Nice, after guiding them to a seventh-place finish last season.
“Yes, like any player, when we do this job we want to touch the very highest level,” Vieira told RMC, when asked about managing Arsenal one day.
“But the very high level I can touch with Nice from the moment we give ourselves the means to succeed and from the moment the way which we work is quite consistent.
“But we do this job, and I do this job, to one day to be the coach of a team that participates in the Champions League.”
The World cup winner, also praised his former manager Arsene Wenger, who recently hailed him for constructing his career ‘intelligently and patiently’.
“I’m really privileged to have worked under Arsene and him saying that is really flattering," he said.
“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.”