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That will allow the 126-year-old retailer to keep its doors open, people familiar with the matter told Reuters.
Lampert raised his bid from an earlier $5 billion offer.
This comes after weeks of back-and-forth deliberations that culminated in a days-long bankruptcy auction held behind closed doors.
Lampert made the proposal through his hedge fund ESL Investments.
It will save up to 45,000 jobs and keep 425 stores open across the United States.
Lampert boosted his bid by adding more cash and assuming more liabilities, the sources said.
The deal concluded in the early morning hours of Wednesday.
There remains a chance the deal could fall apart, as it still must be documented and approved by a U.S. bankruptcy judge. A hearing is expected to be scheduled for later this week.
There are creditors objecting to the deal, one of the sources said.
Spokespeople for Sears and ESL did not immediately return requests for comment.