HARTFORD (AP) — Anthem Blue Cross Blue Shield of Connecticut on January 13 extended its deadline for customers who signed up for coverage through the state’s health insurance marketplace to make their first premium payment in order to receive coverage retroactive to January 1. The new deadline is January 31, the insurer announced, blaming a backlog in processing.

‘‘We thank you for selecting Anthem and apologize for your inconvenience as we continue to work to process all January applications,’’ the insurer said in a message to customers. Anthem said bills will continue to be processed and mailed over the next week.

Anthem said some customers who signed up by the December 23 deadline to receive coverage starting January 1 have not yet received their billing and enrollment materials. Some customers have also experienced long wait times for Anthem’s call centers.

Attorney General George Jepsen’s office has gotten complaints and is looking into them, spokeswoman Jaclyn Falkowski said.

Ms Falkowski said the office has been in communication with Anthem, the state’s Department of Insurance its insurance marketplace, and Access Health CT about the situation.

‘‘We are working to determine why the company is experiencing problems and if there is any way for us to assist consumers in this situation,’’ she said.

The other two private insurance companies offering plans under Access Health CT have extended their premium deadlines. But Kathleen Tallarita, a spokeswoman for Access Health CT, said the bulk of the complaints they have received have been about Anthem, which has more enrollees than the other two firms.

As for the other two, the nonprofit HealthyCT will accept premium payments postmarked on or before Friday, January 17, for coverage retroactive to January 1, while ConnectiCare’s new payment deadline was Wednesday, January 15.

Ms Tallarita said Access Health CT was sending an e-mail January 13 to consumers who signed up with Anthem, informing them of the extended payment deadline and other information. The Department of Insurance also issued a consumer alert, notifying people that Anthem is offering customers various ways to make their payments, including online, by phone, in person at the Wallingford office, and by mail.

‘‘The department of aware that delays in confirmation are causing frustration and uncertainty and has been working with Anthem to ensure that they meet the needs of their customers as swiftly as possible,’’ the state agency said, adding that it requires carriers to make good on their obligations to policyholders.