The regulator wants to prevent another mis-selling scandal by forcing lenders to check whether a customer is eligible to claim under each policy and then to tell them.
About one in five UK adults has a packaged account, a form of current account which can cost up to £300 a year and comes with other products such as ticket discounts or protection from identity theft.
Sheila Nicoll, FSA director of policy, said: “We are concerned that it may be too easy at the moment for firms to sell customers something they do not understand or need”
The regulator has published a consultation paper which also suggests banks and building societies provide customers with an annual eligibility statement to see if their circumstances have changed and tell them if a packaged account is unsuitable.
A spokesman for the British Bankers’ Association said packaged accounts could be a “good deal” but added: “Customers should shop around for the deal which suits their own needs best.”