The watchdog made the statement following the publication of an interim report into the pricing of home and motor insurance.

The FCA said if those customers paying high premiums paid the average price for their risk they could save around £1.2 billion a year.

It is considering how to improve competition, which could include banning or restricting practices such as raising prices for consumers who renew year on year or requiring firms to automatically move consumers to cheaper equivalent deals.

Other changes which have been suggested include restricting the way that firms use automatic renewal and requiring firms to publish information about price differences between their customers.

Christopher Woolard, executive director of strategy and competition at the FCA, said: “This market is not working well for all consumers. While a large number of people shop around, many loyal customers are not getting a good deal. We believe this affects around six million consumers.

“We have set out a package of potential remedies to ensure these markets are truly competitive and address the problems we have uncovered. We expect the industry to work with us as we do so.”

The FCA intends to publish a final report in 2020. The authority worries that consumers who do not switch or negotiate with their provider end up paying high prices.

The FCA discovered that insurers often sell policies at a discount to new customers and increase premiums when customers renew, targeting increases at those less likely to switch. It also found longstanding customers pay more on average.

Gillian Guy, chief executive of Citizens Advice, said: “Last year we submitted a super-complaint showing loyal customers are being penalised hundreds of millions of pounds a year on their home insurance alone. It’s great to see the FCA acknowledging that the insurance market isn’t working for consumers and pledging to crack down on the loyalty penalty.”

The Association of British Insurers has also said the industry will work constructively with the FCA to resolve the issue.