Lampert is competing against a number of liquidators and other investors who want to buy Sears piecemeal — a smaller number of stores or one or two business units, like its home services division. All bidders will compete in an auction that's scheduled for Jan. 14.

Lampert has offered two options to try to keep the company afloat, a $4.4 billion plan that would save 425 locations and a less-ambitious proposal that would keep at least 250 of the little under 700 it had when it filed for bankruptcy in October. The primary bid, made through an affiliate of his hedge fund ESL investments, Transform Holdco, would also save all of Sears' business units, according to an outline of the plan filed with the Securities and Exchange Commission on Wednesday. Lampert's smaller bid would buy a few other pieces of Sears. The rest of the company would likely be liquidated.

Even Sears' biggest advocate, Chairman Eddie Lampert knows his $4.4 billion bid for Sears may not be enough to keep the retailer alive.

His offer for 425 stores is the only one that would keep the company and its brands alive in their entirety, according to people familiar with the matter. His bid for at least 250 stores could keep at least part of Sears running, but those stores would face steep odds in competing with retail titans like Walmart, which has more than 5,000 locations in the U.S. alone.

The downsized offer, therefore, could also ultimately lead to liquidation, even if Lampert's goal is salvation.

The backup offer highlights the continued uncertainty surrounding Lampert's ability to keep Sears alive. Doing so would preserve roughly 50,000 jobs. It would also be small public redemption for the hedge fund investor turned retailer, who has been lambasted by the public for the way he ran Sears after buying it and merging it with Kmart in 2005.

But Lampert's $4.4 billion bid has already faced pushback from the company's unsecured creditors, a person familiar with the situation told CNBC. Sears' restructuring committee is meeting Wednesday at the offices of Sears' legal advisor, Weil, Gotshal & Manges, to discuss whether they will deem Lampert's offer to buy 425 stores "qualified," or financially secure, another person told CNBC.

A likely linchpin is the $1.8 billion Lampert expects to put toward his offer by forgiving debt owed to ESL through a so-called credit bid. Sears' unsecured creditors said last month that they will object to a credit bid. Those creditors believe there may be claims against Sears for transactions under Lampert's leadership. Those deals include Sears' spinoff of Lands' End and transactions with Seritage Growth Properties, a real estate investment trust Lampert created through some Sears' properties.

The company's unsecured creditors are the least protected in a bankruptcy, but could opt to pursue litigation against Lampert for his transactions as way to protest his credit bid. Such a process could take months, or even years — a daunting prospect for a company quickly running out of money. The retailer recently reported a net loss of $950 million for the 13 weeks that ended Nov. 3.

In ESL's offer letter for Sears, it requested that any challenges to its credit bid be dealt with during or prior to a sale process. It also said it is making its bid on the condition it does not need to backstop its offer with cash.

Meantime, ESL has stressed that all transactions it did with Sears during Lampert's tenure were approved by Sears' board.

Lampert's plan to save Sears is more modest than the initial proposal he outlined last month. Lampert's original goal was to buy roughly 500 stores for $4.6 billion. Sears, which filed for bankruptcy in October, has since announced several rounds of store closures that bring store footprint closer to 400 — a sign of Sears' doubt in the viability of Lampert's original offer.

Supporting Lampert's bid is $1.3 billion in credit from Bank of America, Citigroup, and Royal Bank of Canada, another person familiar with the situation tells CNBC. RBC was not previously a lender to Sears.

The people declined to be named because the information is confidential. Sears did not immediately respond to requests for comment. ESL said in a statement provided to CNBC, "We believe that our going concern bid provides the best path forward for the company, the best option to save tens of thousands of jobs and is superior for all of Sears' stakeholders to the alternative of a complete liquidation."