HOFFMAN ESTATES, Ill.—Sears Holdings Corp. reached a last-minute agreement on Tuesday with its chairman and affiliates of his ESL Investments hedge fund that should allow the iconic retailer to avoid liquidation, according to several reports. Sears is considering the new offer from chairman Edward Lampert and his investment funds, but the chairman must officially submit the revised offer, along with a $120 million deposit, by 4 p.m. Wednesday (today) to the New York bankruptcy court, according to the reports.

Upon meeting these conditions, Sears will permit Lampert to participate in the scheduled court-organized auction on Monday, Jan. 14. This is an auction at which various liquidators also are expected to submit their bids for Sears’ operations. After an auction winner has been determined, a deal will require approval from the bankruptcy court on Jan. 31, according to the reports.

Some Sears stores include a Sears Optical department, which are operated by Luxottica Retail. The number of Sears Optical departments has declined over the past couple years as Sears has moved to close stores.

Lampert made a $4.4 billion bid to takeover Sears in late December, however, the parties in the bankruptcy proceedings disagreed over expenses related to Sears during the bankruptcy proceedings.

On Tuesday, Sears had planned to tell the bankruptcy court it had rejected Lampert's $4.4 billion offer to buy the retailer, according to the reports. Sears filed for Chapter 11 bankruptcy protection in October, but Lampert has been looking for ways to keep about 400 Sears and Kmart stores open.