Borders Group Inc
Books-A-Million expressed interest in less than 50 of the book-seller's stores, said the sources, who asked to remain anonymous because Borders' sale process is private.
It is unclear if Books-A-Million has made an official bid or if the talks will ultimately lead to a deal, the sources said.
Borders was set to be sold to a group of liquidators led by Hilco Merchant Resources if no company offered a bid to buy its business by a Sunday deadline. No party offered a bid for the full chain, meaning the bulk of Borders' business is still likely headed for liquidation, the sources said.
A Borders spokeswoman declined to comment on Monday, while a representative for Birmingham, Alabama-based Books-A-Million did not return a call.
Interest in some stores from Books-A-Million, which bills itself as the third-largest U.S. bookstore chain, represents hope for at least a portion of Borders' roughly 400 stores and 11,000 employees.
Borders also has the option of accepting late bids, and could go ahead with an auction slated for Tuesday despite its current lack of participants, in hopes that a last-minute buyer emerges.
As of Monday afternoon, the auction was still scheduled.
Borders, which helped pioneer the concept of book superstores, went bankrupt in February after struggling for years to compete with larger rival Barnes & Noble Inc
Stores not acquired by a going-concern bidder will be sold to the Hilco group, which became the default or stalking horse bidder after a possible Borders sale to buyout firm Najafi Cos, which owns the Book-of-the-Month Club, fell apart last week.
Borders' fate will likely be determined by Thursday, when it is scheduled to seek U.S. Bankruptcy Court approval of what will either be a sale plan, a liquidation plan, or a combination of the two.
Books-A-Million operates about 200 superstores and 30 smaller locations in malls, according to its 2011 annual report. It employs about 5,300 people, roughly half of them full-time, according to the report.
Most of Books-A-Million's superstores include in-store cafes, which it owns and operates under its Joe Muggs brand.
In May, Borders announced it would end a cafe partnership with Starbucks Corp
The case is In re Borders Group Inc, U.S. Bankruptcy Court, Southern District of New York, No. 11-10614.
(Reporting by Nick Brown; editing by John Wallace)