Tax-free Individual Savings Accounts (Isas) were introduced in the UK 11 years ago to encourage people to save.
The average interest rate stands at 0.41%, Consumer Focus said, with many banks and building societies "baiting" customers with short-term higher rates.
The British Bankers' Association (BBA) has criticised the watchdog's approach.
Consumer Focus has submitted a super-complaint to the trading regulator about the £158bn cash Isa market.
Consumer Focus believes that savers are missing out on interest of up to £3bn of interest a year, partly because it is difficult to switch providers.