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Banks have been condemned for "weak excuses" over customers' savings as MPs step up pressure on providers to pass on higher interest rates.
Harriett Baldwin, who chairs the Treasury Select Committee, said: "The time for weak excuses is over."
The financial regulator also said it would soon look at whether minimal returns for loyal savers were offering fair value.
Banks said they were providing a range of deals at competitive rates.
MPs on the committee have been pressing banks and building societies to give a better deal for savers as interest rates have risen, through a series of hearings and letters.
"If the high street banks continue to pay poor savings rates on their instant access accounts, they should make sure their customers know that better rates are available," Ms Baldwin said, describing the subject as of "utmost importance" to the committee.
Lenders stressed that accounts for regular savers, or which locked money in for a set period of time, gave much better returns than instant-access products.
Customers were being alerted to these deals offering higher rates, they added.
Previous figures from the Bank of England show that billpayers dipped into bank and savings accounts at a record level in May. There was £4.6bn more withdrawn than paid into bank and building society accounts.
That was the highest level seen since comparable records began 26 years ago.