Wenger, who has managed the Gunners for more than 20 years but is out of contract at the end of the season, is facing calls for him to step down as the London club endure a turbulent time on the pitch.
They were knocked out of the Champions League at the last 16 stage by Bayern Munich and are currently out of the top four of the Premier League.
Reports suggest that Wenger will sign a new deal at the Emirates Stadium and Merson feels Vieira, who captained the club to titles under Wenger, is the best man to succeed his French compatriot.
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Vieira, who had a spell on the coaching staff of Manchester City after he retired in 2011, is currently coaching New York City in the MLS.
Merson told Sky Sports: “My instincts tell me Wenger is going to sign and he’s waiting for a big win and he’ll sign for another year.
“I wouldn’t be surprised if he stays a year and then Patrick Vieira comes in.
“He’s learning his trade over at New York City, he’s been at Man City, and he brings a winning mentality, he won’t take any rubbish.
“Arsenal look too nice at the moment. I think that could happen and the fans would love that.
“The players don’t seem like they’re working that hard for the manager at the moment, which is sad.”
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