ESL Investments, the hedge fund led by Sears Holdings CEO Eddie Lampert, is proposing to buy the Kenmore brand, Sears' Home Improvement business, its PartsDirect division and some of the chain's real estate.

The retailer saw its sales drop nearly 30 percent during the holiday quarter. Heavily in debt, Sears has been looking for ways to improve its cash flow, including shedding under-performing stores and landing new financing, much from ESL.

In a letter to Sears' board on Friday, ESL said: "We continue to see value in Sears and its underlying assets and believe strongly that with an appropriate runway Sears will be able to complete its transformation to respond to the challenging retail environment."

The Florida-based firm said it believes the value of Sears' businesses are not being reflected in the capital markets. Sears has struggled to find interested buyers, except for its sale of Craftsman to Stanley Black & Decker in 2017.

Sears confirmed receipt of ESL's letter Monday morning, adding the proposal would be reviewed by an independent board of directors. The companies said Lampert, along with ESL President Kunal Kamlani, wouldn't participate in any discussions, negotiations or decisions "except to the extent specifically requested by that committee."

Sears' stock gained 3.8 percent Monday morning on the news. Earlier, it surged 8 percent.

ESL, Sears' second-largest shareholder behind Lampert himself, called Kenmore an "iconic brand" and said it would be prepared to close a deal for this asset within 90 days. The appliance brand recently started selling on Amazon.com.

With respect to Sears' Home Improvement and PartsDirect businesses, Lampert's hedge fund said it valued those assets together at $500 million and would pay for them in cash.

ESL also said it would "be open to making an offer" for Sears' real estate, including the assumption of $1.2 billion in debt obligations. The firm said the expectation in this deal would be for Sears to continue operating its stores, leasing back from ESL.

"In our view, pursuing these divestitures ... will provide an important source of liquidity to Sears and could avoid any deterioration in the value of such assets," ESL said.

Following its divestiture of Craftsman last year, Sears confirmed it's been looking to spin off some of its other assets. The department store chain was recently seen bolstering its Home Services business, hiring more employees within that division.

Sears has meanwhile been selling off its unprofitable stores as another way to raise cash.

Earlier this year, Sears announced a round of more than 100 store closures under both the Sears and Kmart banners, all expected to be completed by the end of this month. The company also recently raised enough money to pay $407 million toward its pension plan in order to allow for the sale of 140 other properties.

Those properties are believed by real estate analysts to be the next Sears locations on the chopping block. Sears hasn't released a list of those stores and hasn't said when those sales might take place. The department store chain is also in the process of selling 16 other stores online.

In response to ESL, Sears said there could be no guarantee a transaction would occur. The company won't comment further "until it determines that additional disclosure is appropriate."

Sears' stock has fallen more than 70 percent from a year ago.

Here's ESL's letter to Sears' board: