The chief executive of Sears is its largest shareholder and a major lender to the company. Now, he wants to be its savior, after the retailer tried — unsuccessfully — to find buyers for some of its more valuable brands.

The chief executive, Edward S. Lampert, said the hedge fund he controls would be willing to buy Sears’s real estate holdings, its appliance-parts business and the Kenmore appliances brand, among other assets. The proposed deal, outlined by Mr. Lampert to the Sears board in a letter released Monday, would infuse the company with new funds as it tries to pay down billions in debt.

“We continue to see the value in Sears and its underlying assets,” Mr. Lampert said in the letter, “and believe strongly that with an appropriate runway Sears will be able to complete its transformation.”

The offer places Mr. Lampert in an unusual position. In proposing that his hedge fund, ESL Investments, buy some of Sears’s best assets, Mr. Lampert would find himself with a stake in both sides of the deal. Mr. Lampert said he would recuse himself from any board discussions about the transaction, which would require the approval of Sears’s shareholders who are not affiliated with ESL.