Edward Lampert speaks during a news conference to announce the merger of Kmart and Sears in New York, Nov. 17, 2004.

ESL's advisors worked through the weekend and into Monday. So far, Lampert's bid is insufficient. If he doesn't agree to concessions, the company will liquidate.

Sears continues to inch toward liquidation, even while leaving Lampert room to have a last-minute about-face. Sears advisors this past Friday had told the bankruptcy judge it planned to announce a liquidation on Monday, one of the people said. By Sunday afternoon, Sears had decided it would give ESL until Tuesday morning before coming to and announcing a conclusion.

Sears and its chairman, Eddie Lampert , have been so far unable to resolve disagreements over his $4.4 billion bid to save Sears and 50,000 jobs by buying it out of bankruptcy through his hedge fund ESL Investments, people familiar with the situation tell CNBC.

A continuing issue is the $1.8 billion that Lampert put toward his offer by forgiving debt owed to ESL through a so-called credit bid. The restructuring committee advising Sears is not confident the bankruptcy judge will allow Lampert to use a credit bid without addressing an ongoing investigation of Sears' transactions under Lampert's management, people have said. Up until Sears filed for bankruptcy, Lampert was CEO with a controlling ownership stake of 31 percent, according to FactSet. His hedge fund ESL Investments owned about 19 percent.

Lampert, who has since resigned the CEO post, has been, so far, unwilling to backstop the bid with his own cash, people familiar said.

Meantime, the company's offer is short of covering the fees and vendor payments it owes, making it "administratively insolvent." Administrative insolvency became a hot-button topic in the restructuring community after Toys R Us' bankruptcy and pursuant liquidation. The toy retailer was forced to strike a deal with the creditors and toy companies it owed after it was unable to cover its expenses in liquidation. The agreement staved off "protracted and expensive litigation.

Sears had hoped that its promise of saving jobs and an American icon would be sufficient to make up for its inadequacies, one of the people said.

For Lampert's bid to pass muster, Sears advisors must decide it is financially viable and the bankruptcy judge must grant it court approval. But the judge's ability to do so is hampered by the overhang from potential litigation.

Sears' unsecured creditors have said there may be claims against Sears for deals done under his tenure as CEO and its largest shareholder, which include Sears' spinoff of Lands' End in 2014 and transactions with Seritage Growth Properties, a real estate investment trust Lampert created through some Sears' properties a year later.

The bankruptcy judge, may, therefore, push to allow a full investigation and, if need may be, litigation for those claims. Such litigation 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.

The people requested anonymity because the information is confidential. Sears and ESL declined to comment.

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