Charging loyal customers more for their insurance could be banned under measures to tackle high home and car policy costs.
The idea is one being considered by the Financial Conduct Authority, which says that six million policyholders pay high prices and are not getting a good deal on their insurance.
Firms could also be required to automatically move customers to cheaper equivalent deals.
Christopher Woolard, executive director of strategy and competition at the FCA, said the market is not working well for all consumers.
He added: "While a large number of people shop around, many loyal customers are not getting a good deal.
"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."
Other ideas proposed in the FCA's interim report on its market study into the sector include stopping practices that can discourage switching, including restricting ways that firms use automatic renewal.
It also suggested firms could be more transparent in their dealings with customers.
The FCA said it was considering whether insurance firms should publish information about price differentials between customers.
The watchdog said insurers often sell discounted policies to new customers and increase premiums when customers renew, targeting those less likely to switch.
One in three people who paid high premiums were vulnerable, the FCA said, and some were less likely to understand insurance or the impact that renewing has on their premium.
The FCA also said that most firms, when setting a price, include their expectations of whether a customer will switch or pay an increased price, something not made clear to the customer.
Gillian Guy, chief executive of Citizens Advice, welcomed the suggestions, saying: "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.
"At the moment these are just proposals. The FCA must now follow through on these bold ideas to stop loyal insurance customers being penalised."
Huw Evans, director general of the Association of British Insurers, said: "We welcome today's report from the FCA, and the industry will continue to work constructively with the regulator to ensure that the market works better for customers.
"It is important that any unintended consequences are carefully considered to ensure that a fair and balanced approach is achieved for all customers.
"Millions of insurance customers get extremely good deals by shopping around regularly, but we agree that the household and motor insurance markets could work better for consumers who do not shop around at renewal.
"This is not an issue unique to insurance, but we are the only sector to have taken voluntary steps to address the issue and these are bearing fruit already."
The FCA will publish a final report and consultation on its ideas in the first quarter of next year.