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.
"We understand that Sears has marketed certain of these assets for almost two years but, with the exception of the Craftsman divestiture, has been unable to reach agreement with potential purchasers on acceptable terms", the letter, posted on Sears' website Monday, said.
ESL offered to submit a bid for the Kenmore brand "i$3 f Sears believes it would be helpful".
Not surprisingly, perhaps, ESL suggests that because Sears has been unable to find a buyer for these assets "on acceptable terms", ESL believes Sears "should aggressively pursue a divestiture of all or a portion of Kenmore, SHIP, and Parts Direct".
Sears has been struggling for years, losing $10.8 billion since 2010, its last profitable year.
Sears stock traded up as much as 12% in Monday's premarket and traded up around 8% shortly after the opening bell at $3.25, in a 52-week range of $1.99 to $13.94.
Juventus and Napoli are set for a wild Serie A finish
Napoli's most hard game left may be their next away trip to Fiorentina, after which they face Torino, Sampdoria and Crotone. Elsewhere, Lazio and Inter kept their push for Champions League football alive on Sunday.
Expressing interest in full acquisition, ESL valued Sears' Home Improvement and PartsDirect businesses at $500 million.
ESL said it also would be open to making an offer for Sears' real estate, including the assumption of $1.2 billion in debt.
Lampert, who combined Sears and Kmart in 2005 after helping bring Kmart out of bankruptcy, has long pledged to turn the company's fortunes around. "This will enable Sears to improve its debt profile and liquidity position, creating the runway to help continue its transformation, and allow these businesses to unlock their considerable potential by further expanding their presence in the marketplace".
Sears said it will review the letter but would not make an additional comment.
Some have argued the strategy will give customers fewer reasons to visit the store.
And while the company still has a "meaningful" amount of real estate and other brands that could provide a source of cash, the more it sells, the fewer assets it will have to fund future shortfalls, Moody's Investors Service said.