CHICAGO — Sears Holdings Corp. chairman Eddie Lampert won out in a bankruptcy auction for the U.S. department store operator with an improved takeover bid of roughly $5.2 billion, allowing the 126-year-old retailer to keep its doors open, The Wall Street Journal and other published sources reported Wednesday.

Lampert’s bid, upped from an earlier $5 billion offer, prevailed after weeks of back-and-forth deliberations that culminated in a days-long bankruptcy auction held behind closed doors. The billionaire’s proposal, made through his hedge fund ESL Investments Inc., will save up to 45,000 jobs and keep about 400 Sears and Kmart stores open across the United States.

The offer beat out a bid, which was supported by most Sears creditors and landlords, by Abacus Advisory Group to close all the stores and sell the inventory.

Lampert boosted his bid by adding more cash and assuming more liabilities, the sources said. The auction, held at the Manhattan offices of Weil, Gotshal & Manges, the law firm representing Sears, 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.

A group of creditors is objecting to the deal, one of the sources said.

Lampert, Sears’ biggest lender and shareholder before it filed for bankruptcy, merged the department store with discount chain Kmart in 2005 in an $11 billion deal.