The City regulator said it had opened a consultation after it found lenders “take advantage” of customers who may not move their accounts to banks that provide higher rates of interest.
Christopher Woolard, executive director of strategy and competition at the FCA, said: “Providers can take advantage of high levels of customer inaction to pay lower interest rates to longstanding customers.
“While many customers have valid reasons for not shopping around, providers must still treat them fairly, while maintaining competitive rates for those who do.
“Efforts to encourage customers to switch have had limited impact and we remain concerned about the way firms are treating customers.
“This is why we are considering the introduction of a basic savings rate for older accounts, which would promote competition and help get customers a better rate of interest.”
Many customers who are willing to shop around for higher rates of return are usually rewarded by lenders but older accounts can suffer, with some banks paying just 0.05% on sums held for a year.
The FCA suggests a basic savings rate (BSR) could result in up to £480m more per year for savers.
Sally Francis-Miles, money spokesperson at MoneySuperMarket, said: “The basic savings rate suggestion seems a good idea in principle, but if a bank is able to set it low, the same problem the FCA is trying to fix – people not switching poor paying accounts for better ones – remains.
“To get top savings rates, you have to keep on top of your own rate and what else is happening in the market.”
Peter Tyler, director of conduct and savings policy at trade association UK Finance, said: “The industry has implemented a number of remedies to improve competition in the cash savings market, helping savers to shop around and find the best deal.
“These include communicating more clearly with customers about the rates they receive, faster Cash ISA transfers and enhanced customer prompts before a rate is reduced.”