Millions of savers who leave money in the same account for years should get a better deal under plans published by the City regulator.
Under the proposals, banks would have to set a single long-term interest rate for all easy access savings accounts to avoid the loyalty penalty.
The Financial Conduct Authority (FCA) said providers could still offer one-year introductory offers.
But the new single rate should avoid banks whittling down rates after that.
Savers who leave their money in the same account with one of the major providers may currently receive as little as 0.1% in interest. Such low rates apply to instant access savings accounts, as well as easy access cash Individual Savings Accounts (Isas).
The best rates available now offer around 1.4%. So that is a difference of £130 a year in interest on a savings pot of £10,000 between the best and worst - even at a time of low interest rates.
A total of 40 million people in the UK have easy access savings, but only 10% of them - primarily pensioners with large life savings pots - move them around to different accounts to make the most of the best interest rates on the market.
They are benefiting from so-called teaser rates, introductory offers made by providers to try to attract new business.
About 20% of savers move their money around occasionally, while 70% leave it in the same account for years. They lose out because banks gradually reduce those rates over time, meaning the interest on their savings may not even keep pace with the rising cost of living.
Christopher Woolard, from the FCA, said the new proposals were designed to offer some protection to these savers and to make offers easier to compare between banks and building societies.
He said the regulator needed to consider the reality for many families, that switching savings accounts was low down on any list of priorities.
Under the plans, firms would be required to pay long-standing customers the same rate as customers who had recently come off an introductory offer.
This Single Easy Access Rate (SEAR) can be changed, but would be the same across all accounts that the bank or building society provides. There would be one SEAR for easy access savings accounts and one for easy access Isas.
At present, for example, one provider has 82 different easy access accounts, with a range of different rates.
The idea, first suggested by the FCA in July 2018, would take effect by early next year if approved following consultation.
'Vote with your feet'
"This [plan] will prevent firms from gradually reducing interest rates over time and make them compete for all their customers," said Mr Woolard.
"The new rate will also make it easier for savers to know whether they are getting a good deal after any introductory offer has expired."
He said banks would compete on their SEAR rate, benefiting existing savers by an estimated £260m, or possibly more if interest rates were to rise. The FCA suggested this could mean £25 extra for the typical saver.
However, UK Finance, which represents the major banks, said that regulation that adds costs to banks would have to be recovered through higher rates for loans and mortgages.
Anna Bowes, co-founder of independent comparison service Savings Champion, said the plans could ignite some competition in the older account savings market, while allowing providers to continue to compete for new savers and possibly encourage switching.
"But this is the problem. The big banks who dominate the savings market pay some of the worse rates on their live accounts, yet savers still deposit funds with them. It is tricky to see how the SEAR could help savers with cash in the High Street banks, in the current market," she said.
"Savers need to vote with their feet to really improve the interest they are earning."