In a letter to the special committee earlier this year, Lampert asked the board to reach out to Sears' debtholders to assess their views on restructuring.
It's still unclear if debtholders would agree to ESL's latest proposal, but the hedge fund said Monday that it would be best to accomplish these things "as a going concern, rather than alternatives that would substantially reduce, if not completely eliminate, value for shareholders."
ESL's goal is to have the department store chain refinance $1.1 billion of debt coming due within the next two years. Sears has about $5.5 billion in debt today.
"ESL's proposal is designed to help create sufficient runway for Sears Holdings to continue its transformation and return to profitability for the benefit of its many stakeholders," ESL President Kunal S. Kamlani said in an email to CNBC.
He said the board should either work with the hedge fund or "offer reasonable alternatives."
Sears earlier this month reported a net loss of $508 million for the quarter, as sales tumbled by a double-digit percentage. Its adjusted loss before interest, taxes, depreciation and amortization widened to $112 million, compared with a loss of $66 million during the same quarter a year prior. Sales at stores open at least 12 months also fell 3.9 percent during the second quarter.
At that time, Lampert rang the alarm bells about potential dire outcomes should the company not restructure its debt load or secure special committee approval for looming asset sales.
"Given the pace and the results so far from our efforts to monetize assets, it is imperative that the company reduce debt, adjust its debt maturity profile and eliminate the associated cash interest obligations," wrote Lampert. "We continue to believe that it is in the best interests of all our stakeholders to accomplish this as a going concern, rather than alternatives that could result in significant reductions in value."
Sears appointed a special committee earlier this year to balance out the potential conflict of interest inherent in ESL's bid for Kenmore and the home improvement business. The committee has frustrated Lampert with its slow pace, a source familiar with the situation has told CNBC.
The committee has been talking to other potential buyers to see if it can fetch a higher offer, that source added.
The special committee's conundrum is amplified by the predicament faced by the Pension Benefit Guaranty Corp., the federal government's oversight organization that guarantees individuals' pensions and has a lien on Kenmore's intellectual property.
Sears shares, which years ago traded above $140, recently hit an all-time low of $1.07. The stock initially soared more than 20 percent Monday morning but was recently down more than 7 percent at around $1.18. Sears' market cap is just about $131 million today.