NEW YORK — Sears’ biggest shareholder has suggested the company sell its well-known Kenmore brand and some real estate holdings, offering itself as a potential buyer.

The ailing company has sold off other major brands as it struggles to stay afloat, with Kenmore a notable remainder of the powerhouse retailer that survived two world wars and the Great Depression.

The private equity firm ESL Investment, headed by Sears chairman and CEO Edward Lampert, said it might buy the assets — Kenmore, Sears Home Services’ home improvement business and the company’s Parts Direct business — if the company is willing to sell.