The Frenchman will leave the club at the end of the season, calling time on a 22-year reign in north London.
Arsenal must subsequently find a replacement, and the former Liverpool boss has been heavily linked with the post.
Indeed, Rodgers has plenty of Premier League experience, and took the Reds to second in 2013-14, just two points behind eventual champions Manchester City.
And Merson has called for the 45-year-old to be installed as Wenger's successor.
"I would like Rodgers to succeed Arsene," he told Sky Sports .
"I like him, the way he plays football. Liverpool nearly won the league under him which they hadn't done for some time before and haven't since.
"People say he had Luis Suarez but you've still got to get a tune out of him. He's got experience of managing big sides as well. There's a lot of pressure at Celtic and he knows the Premier League well."
Merson is also a fan of former Barcelona manager Luis Enrique, but has insisted that the club must find someone who can bring stability to the Emirates Stadium.
"Luis Enrique is the other one, with experience and style of football, but they need to move quickly if they want him because Chelsea might be after him soon," he added.
"The big thing for me is longevity. They need to get somebody in who is going to be there for five years because they don't want a similar situation to Manchester United after Sir Alex Ferguson left."