If Sears doesn't accept the $4.4 billion bid, Transform Holdco said it also would bid individually on some assets, including real estate, certain intellectual property and the Sears Home Services business.
It gives Sears a chance to survive, which appeared to be slipping way heading into the hearing.
In making his bid for what remains of Sears, former CEO and billionaire investor Lampert stipulated any deal would put him and ESL ahead of other creditors down the line, and would be financed with more debt than cash.
Sears on Tuesday had planned to tell the bankruptcy court it had rejected Lampert's $4.4 billion offer to buy the retailer, after it fell short of covering its bankruptcy expenses.
US Bankruptcy Judge Robert Drain in the Southern District of NY, who is presiding over the case, could decide to give Lampert more time to improve his bid, the Reuters sources said, adding that a bankruptcy auction for Sears' assets is not due until January 14.
The revised bid is not official, and will be evaluated in an auction set for January 14 that will compete with other bids from liquidators looking to shut it down.
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Lampert said he plans to offer jobs to 50,000 Sears and Kmart employees. But itis still possible that those wishing to shutdown the company will bid more for the assets than Lampert is offering, which wouldset the company on the path of shutdown once again rather than remaining in business.
The deal was reached after days of "virtually round-the-clock negotiations", Sears attorney Ray Schrock told the court.
Sears filed for Chapter 11 bankruptcy in October.
Unsecured creditors have pushed for Sears to liquidate, partially because they contend they will realize a better financial recovery if it does.
"I remember at the beginning of summer, we'd start flipping through the catalog, having to get the penny loafers of the Mary Janes, the black and white shoes", she said.
Lampert had submitted a bid on December 28. That raised concerns in the negotiations since the maneuver might not be allowed in court, the sources said, given ongoing investigations of Lampert's pre-bankruptcy deals, which the hedge fund manager maintains were proper.