Sears Holdings Corp. is considering a proposal from its largest shareholder to buy some of the troubled retailer’s most lucrative businesses, which the investor says are worth more than stock markets believe.

The value of the Kenmore appliance brand, Sears Home Improvement business, and its Parts Direct division aren’t reflected in the company’s stock price, ESL Investments head Eddie Lampert, who’s also the retailer’s CEO, said in a letter to the company’s board. ESL holds 19 percent of the Hoffman Estates, Ill.-based company’s stock.