HOFFMAN ESTATES, Ill. — The hedge fund controlled by Edward Lampert, Sears Holding Corp.’s chairman and its biggest shareholder, has made an offer it valued at $4.6 billion to buy the bankrupt retailer.

Lampert and ESL Investments Inc. are offering a mix of cash, new loans and debt swaps to buy the entire company, rather than the previously floated plan to acquire its most desirable locations. The new bid is designed to head off outright liquidation of Sears, keeping some 500 Sears stores open and preserving about 50,000 jobs, according to a letter from the hedge fund to the Securities and Exchange Commission.

In addition to the roughly 500 stores, ESL’s bid includes the Kenmore and Diehard brands, the auto center and home services units, the Shop Your Way loyalty program, distribution centers, the corporate headquarters and other ancillary businesses.

Earlier this year, ESL offered $500 million for two parts of Sears’ home services unit and $400 million for the Kenmore appliance brand.

ESL’s takeover bid features financing from a variety of sources and a complicated structure not uncommon in bankruptcy auctions, Reuters reported. ESL proposes to raise up to $1.7 billion in cash through a series of maneuvers that include Sears seeking a new loan backed by collateral and issuing new notes.

The hedge fund would also forgive $1.8 billion Sears owes it in exchange for the company’s assets. Additionally, ESL would assume $1.1 billion in Sears liabilities, including those from its gift cards and loyalty programs.

Sears Holdings is the corporate parent of two storied firms that once ranked as the nation’s two largest retailers. Today, it is a penny stock with market capitalization of under $24 million.

At the time it filed for Chapter 11 bankruptcy, in October, the company had 687 Sears and Kmart stores in 49 states plus U.S. territories, and 68,000 employees. Lampert stepped down as chief executive officer following the bankruptcy filling but remained chairman. He also is Sears’ biggest creditor through his hedge fund.

Through ESL, Lampert purchased Sears, Roebuck & Co. and merged it with Kmart in 2005. Since then, he has extended it billions of dollars in short-term loans, including a loan to support it through bankruptcy.

Lampert’s interest in purchasing the business extends a string of transactions in which he is often on both sides. He is chairman of, and a major investor in, Seritage Growth Properties, a real estate investment trust that ranks among Sears’s biggest landlords. Lampert also has made deals to pull cash out of the company. Last year, Sears began selling Kenmore products on Amazon.com, broadening its reach beyond Sears and Kmart stores. It also began selling its DieHard batteries on Amazon and sold its Craftsman brand to Stanley Black & Decker Inc., which has expanded distribution of the tools, lawn and garden equipment to other retailers.

ESL’s bid for Sears Holdings could face competition from outsiders, and Sears’ attorneys have said they analyze bids to maximize value for shareholders.

To its creditors, Sears might be worth more dead than as a hobbled retailer. The real estate could be sold off piecemeal. Bids have come in from retailers and mall owners, and others are reportedly interested in converting stores to non-retail uses. Additionally, other companies have expressed interest in several of Sears’ services businesses. Brands like Kenmore and DieHard still have some value. And liquidators would likely be willing to buy the inventory and run store-closing sales.