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.

"Sears should aggressively pursue a divestiture of all or a portion of Kenmore, Sears Home Improvement and Parts Direct," Lampert wrote in the April 20 letter, which simultaneously expressed Lampert's interest in purchasing them. "Pursuing these divestitures now will demonstrate the value of Sears' portfolio of assets, will provide an important source of liquidity to Sears and could avoid any deterioration in the value of such assets."

Sears said in a statement Monday that the proposal will be reviewed by a panel of independent board members, without input from Lampert or ESL President Kunal S. Kamlani, who is also a Sears director. Any purchase by ESL would have to be approved both by the independent directors and a majority of Sears' other stockholders, Lampert specified.

ESL valued the Home Improvement and Parts Direct portfolios at $500 million and said, if asked, that it would also consider an offer for the company's real estate, including the $1.2 billion in debt secured by the property.

The company's stock surged 4.5 percent to $3.15 in New York trading after the offer was disclosed. That pared its decline so far this year, fueled by speculation about the company's survival, to 12 percent.