The Financial Conduct Authority had, in July, said banks needed to raise rates for saving accounts faster following criticism that lenders were quick to pass on higher Bank of England rates to borrowers but much slower to "savers".
"There is a more competitive savings market now than July - including many easy access accounts paying above 5%," said Sheldon Mills, the FCA's Executive Director of Consumers and Competition.
The BoE's bank rate is currently at 5.25%.
"We want firms to keep prompting customers in lower-paying accounts to move, and we encourage customers to shop around for the best savings deals," said Mills.
Deposits held in fixed-term and notice accounts rose by 17 billion pounds ($21.40 billion) from July to October, the FCA said.
Meanwhile, those held in bank and building society non-interest-bearing accounts and easy-access accounts reduced by 11 billion pounds, it added.
The watchdog said it had scrutinised assessments provided by nine lenders on their lowest-paying, easy-access savings accounts to see if they were providing fair value, under new consumer protection rules known as the 'Consumer Duty' introduced earlier this year.
It did not rule out taking action.
"We will work closely with firms over the next few months to ensure their assessments are fit for purpose," the FCA said.
"We remain committed to taking robust action against firms who cannot demonstrate fair value." ($1 = 0.7943 pounds)
(Reporting by Yadarisa Shabong in Bengaluru and Iain Withers in London; Editing by Savio D'Souza)
