The struggling retailer, which also operates the Kmart chain, has been selling off crown jewels for years, including its Craftsman tool brand as well as Lands’ End (LE)and Sears Canada, to stave off cash shortages amid ongoing and deep sales declines. On Monday, Sears said that Lampert’s ESL Investments, which has a controlling stake in Sears, sent the Sears board a letter last week urging it to consider to sell off Kenmore, Sears’ home-improvement business, and its Parts Direct business. Such a move would help him keep those better assets in the event of a Sears bankruptcy, an event many observers consider very possible.

The news sent Sears’ stock up as investors cheered the prospect of a cash injection and of some resolution to Sears’ efforts to find a way to either sell or license its popular Kenmore brand of appliances, an effort begun two years ago to no avail. (Sears, however, did begin selling the appliance line on Amazon.com last year.) ESL also indicated it could offer to buy Sears’ real estate, including the $1.2 billion in debt associated with it, if the board asked it to. Under that scenario, Sears would then lease the stores to keep running them, as it has been doing for many of the stores it has sold in the last few years. The company has also closed hundreds of stores under the Sears and Kmart banners during that time.

“In our view, 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,” Lampert wrote in the letter dated Friday, but disclosed by the company on Monday.

Sears (SHLD), which is considered a distressed retailer by Moody’s, has seen its share price collapse in recent years on concerns its business is deteriorating much more quickly than the company can fix it. The company has tried to refashion itself as a membership-focused retailer needing fewer and smaller stores, and added some smaller format stores just for its still-strong appliance business. But it has racked up $6.8 billion in losses in the last five fiscal years, and sales have been plummeting far more dramatically than they have for its peers. Comparable sales, which exclude the stores it closed in the prior year, during the holiday quarter fell 15% in their 25th quarter of decline in a row, despite a strong climate for retail sales. Sears shares were trading above $3 on Monday morning, a fraction of their $144 all-time high eleven years ago.

The company said on Monday that a board committee is looking at ESL’s letter.