Sears chairman and former CEO Edward Lampert and his hedge fund ESL Investments raised their bid for beleaguered retailer Sears Holdings Corp. (SHLDQ) by more than $600 million, bringing ESL's total bid for the company above $5 billion.  

ESL submitted a definitive bid proposal through its Transform Holdco subsidiary that includes the assumption of an additional $663 million in liabilities that were tacked on to ESL's initial offer of $4.4 billion. 

Earlier this week, U.S. Bankruptcy Court Judge Robert Drain agreed to give Lampert a Wednesday afternoon deadline to improve his bid ahead of a scheduled January 14 asset auction for whats left of the once mighty retailer. 

Sears has until 5 p.m. Eastern Time January 13 to agree to allow ESL to be a part of the auction. If not, the bid will be pulled by ESL, according to the 13-D filing. The bid will also be pulled if Sears does not confirm that ESL has been selected as a "successful bidder" by 5 p.m. ET on January 16. 

Lampert's offer included a $120 million cash deposit, $17.9 million of which is non-refundable,  Sears lawyer Ray C. Schrock said during a hearing before Drain in suburban New York Tuesday.

The Jan. 14 auction could spell an end to Sears' 126-year history. After years of slow declines, Sears closed even more stores in recent months after filing for Chapter 11 bankruptcy protection in October.

The company's assets and about 68,000 jobs are at stake if the company completely shuts its doors. However, news of the temporary reprieve sent Sears stock up as much as 30% Tuesday to 39 cents a share. 

Sears shares were little changed at 39 cents in Thursday trading. 

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