NEW YORK (Reuters) - Negotiations between Sears Holdings Corp Chairman Eddie Lampert and the bankrupt U.S. department store operator approached a resolution late on Tuesday as the billionaire hedge fund manager faced the choice of improving his $5 billion offer for the company or ending his takeover plans, people familiar with the matter told Reuters.

FILE PHOTO: The Sears headquarters in Hoffman Estates, Illinois, seen in this November 17, 2004 file photo. CPROD REUTERS/John Gress

After two days of haggling, U.S. Bankruptcy Judge Robert Drain set a Wednesday deadline to complete the bankruptcy auction for the 126-year-old retailer, the people said. Sears was weighing Lampert’s offer against the sum it would recoup by winding down its business and selling its assets off in pieces.

Liquidating Sears would end the department store in its current form, meaning layoffs for as many as 68,000 people and the closure of about 500 stores.

Last week Lampert, through his hedge fund ESL Investments Inc, made an improved $5 billion offer for Sears after the company turned away his earlier $4.4 billion bid.

But the same issues that dogged his earlier offer were proving to be hurdles for his $5 billion proposal.

Sears believes the bid falls short of covering the bills the retailer has racked up since filing for bankruptcy protection in October, the people said. A bedrock principle of bankruptcy cases is that those expenses, known as administrative claims, must be fully repaid.

Sears is also pushing Lampert to offer more money in exchange for a legal release that would shield him from future litigation over transactions he did with the company in the past, the people added. Lampert has said the deals were proper.

Lampert’s $5 billion offer had taken on another $600 million-plus in liabilities, and marked $35 million for a so-called release from litigation.

Sears declined to comment.

The bankruptcy auction for Sears began on Monday morning in Manhattan at the offices of law firm Weil, Gotshal & Manges LLP, which is representing the retailer. On Tuesday, representatives for Sears and Lampert briefed Drain on the status of the auction, the people said.

Drain pressed both Lampert and the company to continue negotiating and to end their discussions one way or the other before Tuesday comes to a close, the people said.

Sears last week made liquidation preparations before reaching a last-minute deal with Lampert that resulted in his latest bid, leading to the auction that commenced on Monday.