He said: "Switching platforms should be simple, straightforward and efficient for customers and the whole industry has a responsibility to deliver that.
"Quilter is supporting the FCA’s aim to make moving platforms simpler for customers.
"This includes our support for the proposals to end exit fees, incorporating ‘comparable services’ providers within these rules and requiring receiving platforms to give customers the option to convert into discounted share classes where available."
Mr Levin is also calling on the industry to implement the "best possible in specie conversion process".
He said: "In order to do so we need to agree, as an industry, how it is most appropriate to execute the conversions.
"Enhancements to existing technology used across the industry will be required, but we believe that making fund managers responsible for undertaking the conversion is most likely to produce a good outcome for the customer.
"We ask the industry to come together, with the support of the FCA, to explore this option in detail."
Andy Bell, founder and chief executive of the AJ Bell platform, said different share classes of the same fund acted as a "significant barrier to transfers" because not all platforms can hold all share classes.
He said: "The solution here is to reintroduce cash rebates, on the basis that these must all be paid to the customer’s account and cannot be retained by the platform.
"This would enable there to be a single retail share class for each fund with platforms able to negotiate discounts for their customers in the form of cash rebates that are paid into the customer cash account on the platform.
"I believe that any fears that these cash rebates will be used to confuse platform pricing are unfounded and a single share class per fund would enable easy transfers between platforms."
The Financial Conduct Authority has declined to comment.