There has been much speculation over the future of the Frenchman, even after being handed a two-year contract extension last year, which could see his tenure at the club stretch to 23 years.
Merson, however, would prefer to see the long-serving manager leave this summer, with Rodgers being his top choice to replace him.
"I don't know why he doesn't get the Arsenal job," he told Sky Sports News. "He fits Arsenal great, the nearest Liverpool have ever come to winning the league for a long long time is when Brendan Rodgers was in charge.
"He's done nothing wrong at Celtic. To do what they've done, people can say it's Scottish football and Celtic are head and shoulders above, but every team they played on that run treated it like a cup final, which is difficult sometimes.
"I think he now needs a change and needs to come back down. I'll be shocked if he didn't get a good job, I'd be quite shocked if he didn't get the Arsenal job."
Arsenal face AC Milan at the Emirates in the second leg of the Europa League round-of-16 tie, with an aggregate score of 2-0 in their favour following a solid away performance at San Siro.