The service includes boiler breakdown cover and annual servicing and costs from £14 a month.
Customers are complaining that they have had their boiler maintenance appointments repeatedly cancelled.
Social media posts from customers had claimed that during cold weather they had to turn on costly electric heaters while others remained without power.
Centrica has been warned by the Financial Conduct Authority (FCA).
A spokesperson from the UK’s financial regulatory body told ELN: “We are aware of the issues and are working with the firm to ensure these are resolved as quickly as possible and that customers are treated fairly.”
It has been reported that the company has apologised for the ongoing issues.
ELN has contacted British Gas and Centrica – neither responded before publication.