The Financial Conduct Authority (FCA) is considering whether to intervene in order to boost competition within a market in which six providers hold roughly two-thirds of all cash savings balances in the UK.
It is concerned that because many consumers do not shop around, banks are able to pay lower interest rates to long-standing customers.
The FCA said: "While some consumers may switch away in response to the lowered rates, a significant number of consumers do not."
The investigation has been focused on easy access accounts and no-term cash ISA accounts.
The FCA will now undertake further research before taking a view on whether it should take further steps to ensure competition is working in the interests of consumers.
Kevin Mountford, head of banking at MoneySuperMarket, said that if people did switch their savings accounts more often, this would put more pressure on banks and building societies to offer better deals "across the board".