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Banks could be forced to set a minimum interest rate on their savings accounts, the Financial Conduct Authority (FCA) has suggested.
The FCA said it was concerned that savers who stay with the same bank or building society for a long time get poor returns on their money.
Some banks currently pay just 0.05% a year on instant access accounts.
The Basic Savings Rate (BSR) would apply to all easy access cash ISA products, as well as savings accounts.
It would be applied after the account had been opened for a set period, for example one year.
"Providers can take advantage of high levels of customer inaction to pay lower interest rates to longstanding customers," said Christopher Woolard, executive director of strategy and competition at the FCA.
He said customers who do not shop around for higher rates should be treated fairly by their banks or building societies.
It would be up to each bank to set their own BSR, which would apply across all their instant access accounts. The rates would then be published on the FCA's website, so consumers could compare them easily.
The FCA has tried previously to encourage savers to shop around for better rates, but with limited success.
"Efforts to encourage customers to switch have had limited impact and we remain concerned about the way firms are treating customers," Mr Woolard said.
"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."
The highest returns are generally offered by current accounts, where savers can get up to 5% a year in some casesalthough such accounts have strict limits on the amount of money that can be put in.
The FCA said a Basic Savings Rate (BSR) could enable customers jointly to earn up to £480m a year more than they do at the moment.
It said around a third of accounts were opened more than five years ago.
And on average, customers with such accounts earned 0.82% less than people whose accounts were opened more recently.
Interested parties can respond to the FCA's discussion paper between now and 25 October.