Lampert's $4.4 billion offer does not complete the sale, but rather starts an auction that is due to be completed on January 14.
The reprieve came 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 sought to preserve 425 stores and 50,000 workers.
The deal was reached after days of "virtually round-the-clock negotiations", Sears attorney Ray Schrock told the court.
There is a court hearing Tuesday morning in White Plains federal bankruptcy court, after which Sears is expected to announce whether it will liquidate, according to sources. But many retailers end up going bust after filing, despite plans to stay in business. The bid would have kept 425 stores open and offered jobs to 50,000 of its remaining employees. Store closings continued throughout 2017 and 2018, with Sears Holdings declaring bankruptcy in mid-October 2018.
But it wasn't immediately clear just how much additional cash Lampert will pay for the assets.
Lampert's bid proposed forgiving $1.3 billion of debt he holds in exchange for ownership of the reconstituted Sears, a bankruptcy maneuver known as a credit bid.
Unsecured creditors have pushed for Sears to liquidate, partially because they contend they will realize a better financial recovery if it does. It was not immediately clear Tuesday how much, if any, debt forgiveness will be included in this revised offer.
Sears' lawyers said a revised bid will require Mr Lampert to make a $120m (£94.4m) down payment, through his hedge fund, by 4pm NY time on Wednesday. Those creditors, which include Sears landlords and bondholders, have also questioned Lampert's pre-bankruptcy transactions with the retailer.
Under Lampert, Sears has bought time over the years by spinning off stores and putting on the block the brands that had grown synonymous with the company, such as Craftsman.
Lampert and his hedge fund argue the loans were proper and made to keep Sears alive.