Barclays Bank, which sells insurance underwritten by Norwich Union, may use customers' bank account details to cherry-pick risks.
The bank is understood to be looking at "scoring" potential customers according to their banking records. Those who miss credit card payments, for example, may be offered worse terms for their cover because they are deemed higher risk.
A spokesman said: "Barclays is considering a number of options to make our policies cheaper for customers, of which this is one. Any decision would be strict accordance with the correct government regulations."