Lampert submitted a bid on December 28 and Sears had until Friday to accept it, but the company didn't comment before Tuesday's hearing.
Lawyers for Lampert and his hedge fund, ESL Investments, were pressing for more time to sweeten his offer.
The revised deal was reached after days of "virtually round-the-clock negotiations", Sears attorney Ray Schrock told the court.
The reprieve came Tuesday after what Sears lawyers described to a bankruptcy judge in NY as "round-the-clock" negotiations following the company board's initial rejection of Eddie Lampert's proposal, which would preserve 425 stores and 50,000 workers.
Once the largest retailer in the United States, Sears failed to adapt to changes in consumer spending behavior, particularly the rapid shift to online shopping and the ascent of Amazon, whose e-commerce platform is often compared to the once-ubiquitous Sears mail-order catalog.
By deeming Lampert's revised bid acceptable, Sears started the clock on an auction period that ends on January 14.
"The bankruptcy process is burning through a lot of money and these assets are more valuable today rather than tomorrow", said a source close to the creditors.More news: Oil prices edge up on trade talk hopes and OPEC cuts
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Sears Holdings Corp shares more than doubled after news emerged that Lampert would have another chance to rescue Sears from liquidation.
Lampert's is the only bid created to keep Sears open.
The bid now requires Lampert to deposit $120 million by 4 p.m. Wednesday through his ESL hedge fund.
Sears has not said how many other offers it received or whether any others would let the retailer avoid liquidation. Store closings continued throughout 2017 and 2018, with Sears Holdings declaring bankruptcy in mid-October 2018.
The offer from Lambert is $4.4 billion, however it does not complete the sale.
Lampert touted his merger of Kmart and Sears as laying the groundwork for their revival, a scenario that did not work out, and had the hedge fund manager relinquishing his role as CEO just after Sears filed for bankruptcy protection last fall.
Sears' bankruptcy, which includes discount chain Kmart, followed a decade of revenue declines, hundreds of store closures, and years of deals by Lampert to turn around the company he put together in 2005 through an $11 billion deal.