Sears Chairman Eddie Lampert is working on a higher bid to keep Sears alive, a person familiar with the situation told CNBC.

The bid, though, is not yet finalized and submitted, the person said. When it is, Lampert will be required to publicly file it with federal regulators.

Bloomberg first reported Wednesday afternoon on Lampert's latest efforts to keep Sears alive.

Sears, which also owns Kmart stores, filed for bankruptcy on Oct. 15. If Lampert is unable to cobble together an offer to buy the company out of bankruptcy, it will be pushed into liquidation.

This latest attempt comes one day after Lampert's initial $4.4 billion bid to save Sears was rejected by the company. It faced a number of challenges, including being short of covering Sears' administrative expenses, making it "administratively insolvent." It also relied on $1.8 billion of debt owed to Lampert's hedge fund, ESL Investments, a move that Sears' unsecured creditors objected to.

On Tuesday, though, a bankruptcy judge gave Lampert more time to improve his offer. The judge said ESL would be required to pay a $120 million deposit by 4 p.m. Wednesday. Sears will then hold an auction on Jan. 14, when it will compare Lampert's offer against others by liquidators. Once a bid is accepted, it would still need to be approved by the bankruptcy court.

The structure and size of Lampert's new bid could not be immediately determined. It also wasn't known whether it addressed the original offer's shortcomings. The person speaking to CNBC requested anonymity because the talks were confidential.

A spokesperson for ESL referred CNBC to a statement made Tuesday about Lampert's efforts to keep Sears alive.

"ESL appreciates the encouragement from the court and the constructive engagement of the debtors as we work to formalize our going concern proposal so that it can be evaluated at the upcoming auction. As we have said before, our proposal provides substantially more value to stakeholders than would be the case in liquidation and is the only option to save an iconic American retailer and up to 50,000 jobs. We believe in Sears and will continue to do everything we can to ensure that it has a profitable future," it read.

Sears declined to comment.

— CNBC's Lauren Thomas contributed to this reporting.