The Financial Conduct Authority (FCA) said last year that first-time customers for cash savings products were getting higher interest rates than existing customers when they came off introductory offers, dubbed a loyalty penalty in the industry.
Under the FCA proposals, firms will have to set a single easy access rate (SEAR) across all accounts that let savers withdraw their money when they want in a move that would boost interest payments by 260 million pounds a year.
Banks will still be able to compete by offering an introductory rate that is usually cut after a set period.
"Firms will have flexibility to offer multiple introductory rates for up to 12 months, then they will need to choose one SEAR for their easy access cash savings accounts, and one for their easy access cash savings ISAs," the FCA said.
The FCA said competition in cash savings products was not working well for many of the 40 million consumers who hold either an easy access savings account or easy access cash Individual Savings Account (ISA).
Christopher Woolard, the FCA's head of competition, said the proposed new rule should be in place before the start of the 2021-2022 tax year and will help the 90% of cash investors that switch products infrequently or not at all.
UK Finance, which represents banks, said that regulatory intervention that increases the overall cost of deposit funding for providers will, in general, result in providers having to raise the cost of loans to maintain adequate margins.
Companies will also have to publish data every six months on the SEARs they offer so investors can compare different institutions when they first open a savings account.
One bank was offering over 80 easy access accounts paying between 1.45% and 0.25% interest even though they are virtually identical, Woolard said.
Citizens Advice, a consumer body campaigning against loyalty penalties, said the FCA "must now hold its nerve and make sure these proposals are introduced".
The Bank of England's main interest rate has been hovering around record lows for the past decade, hitting the rates on products such as popular individual savings accounts or ISAs - even before discriminatory rates for long-standing customers.
The watchdog has put out the proposals for public consultation.
"The plans from the regulator would be a dramatic change to the cash savings market and will mean banks can't hide behind a vast array of different interest rates for cash accounts," said Laura Suter, personal finance analyst at investment platform AJ Bell.
"Banks will also no longer be able to quietly ratchet down interest rates they pay on cash savings over time, in the hope that customers won't notice," Suter said.
By Huw Jones