ESL Investments Inc. upped the value of its bid for Sears Holdings Corp. to an "excess" of $5 billion, agreeing to take on additional liabilities and purchase some of Sears' real estate.

In a letter dated Jan. 9 to Sears' investment bank Lazard Freres & Co. LLC and filed with the SEC on Jan. 10, ESL said it would assume up to $663 million in new liabilities under the deal. That amount includes up to $166 million Sears owes on goods it has ordered but not yet received from vendors. ESL is Sears Chairman Eddie Lampert's hedge fund, and it is making its bid through its Transform Holdco entity.

Under the revised proposal, ESL would also purchase approximately 57 additional real estate properties from Sears, though the letter did not specify how many of those properties would be Sears stores. ESL would also acquire inventory with a book value of at least $147 million that Sears has paid for but not yet received as well as assets related to warranties that Sears offers on some of its products.

"We believe our proposal will provide substantially more value to stakeholders than any other option, in particular a liquidation, and is the best path forward for Sears, its associates and the many communities across the United States touched by Sears and Kmart stores," an ESL spokesman said in an email to S&P Global Market Intelligence.

A Sears spokesman did not immediately respond to a request for comment on the revised bid. Multiple news outlets reported Jan. 9 that Lampert was working to increase ESL's bid for Sears after the retailer rejected an earlier offer worth $4.4 billion.

ESL's revised bid will have to face off against any other proposals at an auction slated for Jan. 14.