On Tuesday morning, it looked like Sears Holdings' (NASDAQOTH:SHLDQ) long-promised turnaround had finally run out of time. The company filed for bankruptcy back in October, and a long-shot proposal from Chairman and former CEO Eddie Lampert is the only hope of keeping Sears and Kmart in business. Yet the company was prepared to reject that proposal at a key bankruptcy court hearing in favor of liquidation.

However, a last-minute agreement between Sears and Lampert's ESL Investments hedge fund provides a short window of time for Lampert to try to keep Sears alive. Still, with customers having abandoned the Sears and Kmart brands years ago, this rescue attempt will probably fail.

An offer with some serious holes

Late last month, ESL Investments put in a formal bid for Sears Holdings' assets, following up on an informal proposal presented several weeks earlier. The hedge fund hopes to keep about 425 Sears and Kmart stores operating, while maintaining control of the retailer's real estate and key brands like Kenmore and DieHard.

ESL valued the final offer at $4.4 billion, consisting mainly of new loans from third-party lenders, forgiveness of debt owed to ESL, and the assumption of certain liabilities by the "new" Sears. That may seem like a generous sum, considering the state of Sears and Kmart, but the company hasn't found it very palatable.

There are several potential land mines. First, creditors have accused Lampert of stripping assets from the company and loading up Sears Holdings with secured debt in a way that unfairly benefited ESL. ESL's offer is conditional on the fund (and Lampert) being released of any potential liability related to issues of this sort. That stance has been a big stumbling block.

The exterior of a Sears full-line store.

Image source: Sears Holdings.

Second, ESL's bid doesn't include enough cash to cover all of Sears' vendor payments and legal and advisory fees, raising further litigation risks. Third, there's a good chance that selling off Sears' assets piecemeal would raise more money than what ESL is offering.

The net result is that with the bankruptcy auction looming -- it is scheduled for Jan. 14, less than a week from now -- ESL still hasn't submitted a bid that Sears Holdings finds adequate.

A reversal, just in the nick of time

In light of this impasse, Sears Holdings seemed to be on the verge of asking a bankruptcy court judge for permission to liquidate at the hearing held on Tuesday. However, at the last minute, the company decided to give ESL one more chance to improve its offer.

ESL Investments will be able to participate in the bankruptcy auction next Monday, provided it can come up with a $120 million deposit by 4 p.m. EST on Wednesday. That includes a $17.9 million nonrefundable portion that would fund Sears Holdings' cash burn for the next several days, to appease creditors who have been saying all along that every day of delay reduces the amount of cash that will ultimately be available to repay them.

It's still not certain that Lampert will be able to come up with the $120 million or that ESL will sweeten its bid enough to prevail at the bankruptcy auction. But at least there's still a chance that Sears could survive in some form.

It may not matter

Lampert still faces long odds in his effort to keep some Sears and Kmart stores operating. The company's steep sales declines and massive losses over the past several years prove that the two struggling chains don't really have a place in the American economy going forward. It will be hard for ESL to pay enough to top the alternative of selling Sears Holdings' inventory, real estate, brands, customer data, and non-retail businesses separately.

Even if ESL does win, it will probably only grant Sears and Kmart a temporary reprieve. With little remaining debt and a shoestring budget, perhaps the chains could stay solvent for a year or two. However, at the first sign of economic weakness (if not sooner), they would likely begin hemorrhaging cash again -- leading to their ultimate demise.

Adam Levine-Weinberg has no position in any of the stocks mentioned. The Motley Fool has no position in any of the stocks mentioned. The Motley Fool has a disclosure policy.