A recent report claimed that the champions were lining up a big-money move for the England international next summer.
Marsh, who played for City from 1972 to 1976, thinks that Barkley should be looking to make the switch to the Etihad.
He told talkSPORT: "I would advise Ross Barkley to move to Manchester City in a heartbeat.
"He has the opportunity to become one of the greatest British players we've ever seen.
"He's the sort of player who can move to Manchester City and, unlike others, can get into the first-team as a youngster, dominate and become a truly great footballer."
Barkley is currently out injured with a knee problem.