ESL Investments also made an offer to buy the Home Improvement business of the company's home services division for as much as $80 million in cash, according to the filing with the U.S. Securities and Exchange Commission.
It said the offers were made in a letter to Sears' board.
The offer for Kenmore is conditional on ESL receiving equity financing from a potential partner, according to the filing. No partner was named.
Lampert said in April Sears should sell its Kenmore brand, home improvement businesses and real estate, and that ESL Investments would bid in any sale.
Sears formed a special committee in May to explore the proposal.
In its fiscal first quarter ended May 5, Sears reported a net loss attributable to the company of $424 million, or $3.93 per share, compared with a profit of $245 million, or $2.29 per share, a year earlier. The company's store closures brought down its revenue by nearly 31 percent in the first quarter.