The ex-forward, who played for City between 1972 and 1976, feels that Barkley could develop into a world star at the Etihad Stadium.
The England international, 20, is a target for all three clubs and would reportedly command a transfer in the region of £50 million.
Barkley tipped to become star at City
Manchester City have a track record of snapping up young British talent and failing to hand them first-team opportunities, such as Scott Sinclair and Jack Rodwell, but Marsh believes it’s a different situation with Barkley.
"I would advise Ross Barkley to move to Manchester City in a heartbeat,” he told talkSPORT. "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.
"I would advise anyone at the moment who has the chance of becoming a top player to go to City, it's a wonderful club now. They've got everything in their favour, from their training conditions to being able to play around great players.”
Barkley's training habits revealed
Marsh went on to reveal the amount of hard work Barkley puts in on the training ground - and believes the talented midfielder is capable of becoming Manchester City’s version of Wayne Rooney.
"This morning I spoke with someone who has seen every game Ross Barkley has played, even for Everton's youth team,” Marsh continued. “He told me this kid loves his football, he trains harder than anybody, he stays behind after training and is a wonderful young man as well, and they truly believe he will go on to become a world star.
"Another player Everton let go as a youngster, Wayne Rooney, became the biggest thing in British football when he moved to Manchester United. I put Barkley in the same category.
"Only time will tell but I think this kid can be breathtaking, both for England and Man City, if he decides to move there.”
Marsh: Yaya Toure will go
Marsh also believes that Barkley could slot in and replace Yaya Toure at Manchester City. There are question marks over the Ivory Coast international’s happiness at the Etihad Stadium, and Marsh thinks the Citizens’ star man will leave the club in the near future.
"I think Yaya Toure could well be out of the back door,” he added, “so that's another reason why I think Barkley would be great at City."
Who will sign Barkley?
Selling Toure in order to finance a deal for Barkley would allow Manchester City to add a homegrown player to their ranks, while simultaneously helping them avoid incurring any additional punishment from UEFA for falling foul of their Financial Fair Play regulations.
Chelsea know that they too will need to offload a player before they can purchase Barkley; Jose Mourinho is reportedly willing to sell Ramires, who is wanted by Real Madrid.
Meanwhile, Manchester United spent huge sums of cash on revamping their squad during the summer, but would love to add Barkley to their ranks.
The midfielder, however, recently stated that he is happy working under Roberto Martinez at Goodison Park and subsequently signed a new four-year contract with the Toffees in July.