Sears Chairman Eddie Lampert has gone back to the drawing board in his bid to save the company from liquidation, emerging with a revised package of around $5 billion, according to CNBC.

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."

As part of the new bid, Lampert will assume tax and vendor bills incurred since Sears' October bankruptcy, the person said. As with its previous bid, it will aim to keep roughly 50,000 jobs. -CNBC

It was unclear whether Lampert's bid also relies on debt forgiveness through a so-called "credit bid." The company's creditors cried foul over a $1.8 billion credit bid that had supported Lampert's previous offer.

If Sears deems Lampert's bid acceptable - his company ESL will be allowed to participate in a January 14 auction against other buyers. Lampert's offer is the only one which would keep the company intact.

On Tuesday Sears' advisors told a bankruptcy court that it will evaluate the merits of the credit bid during the January 14 auction.

Whatever Sears decides, any offer would need to be approved by the bankruptcy judge in a January 31 court hearing - who would also have to eventually sign off on a credit bid, if Lampert relies on one.