Around the time Borders Group Inc. was founded four decades ago, two immigrants who would make their fortunes as buyout moguls arrived in the U.S. from the Middle East. Now, the two rivals represent the struggling bookstore chain's best hope of avoiding liquidation.

Jahm Najafi and Alec Gores, who hail from Iran and Israel, respectively, are separately negotiating to buy Borders out of bankruptcy proceedings. The veteran investors, whose private-equity firms hunt for distressed assets, are betting they can scoop up a well-known business on the cheap and turn it around at a profit.

They have followed different paths to become potential white knights for Borders. Mr. Najafi, who owns the Book-of-the-Month Club, is known in publishing and retailing circles but has mostly avoided the limelight. Mr. Gores is one of a trio of Los Angeles brothers who are influential in finance and entertainment, sometimes kicking the tires on a movie studio.

This following account is based on interviews with more than half a dozen people with knowledge of Borders' discussions with the two potential buyers. These people cautioned the talks remain fluid and details could change ahead of an auction of the company, which is set for July 19.

The fate of Borders, the second-largest U.S. bookseller by sales, rides on whether Messrs. Najafi or Gores agree to purchase most of its remaining assets. Absent a buyer, the retailer, which employs more than 11,000 people, will be forced to liquidate. Borders has to name an opening bidder for the bankruptcy-court auction by July 1.

Mr. Gores's and Mr. Najafi's negotiations with Borders' advisers at law firm Kasowitz Benson Torres & Friedman, investment bank Jefferies & Co. and turnaround firm AlixPartners are expected to intensify this week. In recent days, no one has appeared to have gained a clear edge in the jockeying for the rights to make an opening bid.

Borders hopes to fetch between $250 million and $350 million in a sale. It already has closed more than a third of its stores since seeking bankruptcy protection in February, leaving it with about 400 outlets. Since its filing, the company has posted losses of more than $191 million, according to financial reports through May 28.

"I'm equally excited about both partners and enthusiastic that they see the potential and opportunities in the Borders brand," said Borders President Mike Edwards.

A plan by Mr. Gores, 58 years old, would save roughly 250 Borders stores, the bulk of them so-called superstores, which he intends to revamp into more appealing destinations akin to Apple Inc. AAPL -1.57% Apple Inc. U.S.: Nasdaq $523.47 -8.35 -1.57% April 7, 2014 4:00 pm Volume (Delayed 15m) : 10.23M AFTER HOURS $524.40 +0.93 +0.18% April 7, 2014 7:59 pm Volume (Delayed 15m): 123,650 P/E Ratio 12.90 Market Cap $474.68 Billion Dividend Yield 2.33% Rev. per Employee $2,163,820 04/08/14 Samsung Galaxy S5 Review: Wate... 04/07/14 Samsung Forecasts Decline in O... 04/07/14 Microsoft: We're in an 'AI Spr... More quote details and news » 's outlets.

His Gores Group private-equity firm, which manages $4 billion, would also emphasize developing a more robust online business for Borders. The bookseller has effectively turned over its e-book business to Kobo Inc., a Canadian e-retailer in which it has an investment stake.

Mr. Gores is in discussions with more than a dozen companies about teaming up with Borders to showcase products in its stores, in an attempt to make them a destination for more than just books. Hewlett-Packard Co. HPQ +0.18% Hewlett-Packard Co. U.S.: NYSE $32.70 +0.06 +0.18% April 7, 2014 4:00 pm Volume (Delayed 15m) : 12.18M AFTER HOURS $32.82 +0.12 +0.37% April 7, 2014 7:57 pm Volume (Delayed 15m): 17,236 P/E Ratio 11.81 Market Cap $61.86 Billion Dividend Yield 1.78% Rev. per Employee $352,917 04/06/14 SAC Names Bart Schwartz New Co... 04/04/14 H-P Officials Seized in Pakist... 04/03/14 Can GMAC Be More Than a B-Scho... More quote details and news » , for instance, is in talks with Mr. Gores about featuring the new e-reader H-P has under development in Borders' stores, along with other products. Borders, in turn, could offer customers discounts for downloading books from HP's e-reader in stores, among other promotions. H-P declined to comment.

Overall, Mr. Gores would pay around $250 million for the stores and other assets, such as Borders' website and customer lists. The price could rise depending on concessions Mr. Gores receives from publishers and landlords.

Mr. Gores's team wants publishers to give it more normal trade terms of one or two months to pay bills. Borders stopped paying vendor bills before seeking bankruptcy protection in February, and many publishers now demand cash in advance before shipping books. Meanwhile, Borders has asked its landlords for rent concessions that exceed those the company has achieved through bankruptcy proceedings.

In discussions with Borders and its stakeholders, Mr. Gores's team has emphasized its track record. Among Mr. Gores's most successful bets: VeriFone Systems Inc., for which he paid $50 million in 2001.

After a series of transactions, Mr. Gores reaped about $360 million restructuring the provider of electronic-payment technology.

Mr. Najafi, 48, has mostly kept a low profile. He and his brother, both investors in the Phoenix Suns basketball team, made a fortune on real estate. His most successful bet: Network Solutions, purchased for just $20 million plus $80 million in assumed debt.

He later sold the company for $800 million. Unlike many other private-equity firms, Mr. Najafi's doesn't raise money from outside investors.

Publishers key to Borders' survival know Mr. Najafi and his Najafi Cos. from his July 2008 purchase of the Book-of-the-Month Club, the Doubleday Book Club and Columbia House from Bertelsmann AG. He took several IT platforms, call centers and distribution centers and consolidated them into one.

Mr. Najafi could bolster his book clubs by tapping the more than 43 million Borders customers who are members of the chain's rewards programs and put outposts in the chain's stores. But Mr. Najafi doesn't purchase companies based on synergies alone.

While talks remain fluid, Mr. Najafi has expressed interest in roughly 300 or more Borders stores. He believes Borders has a strong brand that retains an attractive base of customers Borders can build on by providing them with additional goods and services, much the way he increased revenue at Network Solutions by offering e-mail and e-commerce capabilities.

Mr. Najafi has a patient attitude toward the company's digital businesses. The Kobo e-reader, though not big in the U.S., has received positive reviews, and Borders could sell other devices later.

Two Would-Be White Knights

A look at two veteran investors involved in competing talks to buy bookstore chain Borders Group Inc. out of bankruptcy.

Alec Gores, 58 years old.

Heads Los Angeles private-equity firm Gores Group.

Has bought more than 80 companies.

Restored Learning Co. to profit 75 days after buying it in 2000.

Bought National Envelope Corp. out of bankruptcy for $208 million in September 2010.

Teamed with brother Tom the same month to buy DVD, CD and videogame distributor Alliance Entertainment.

Jahm Najafi, 48 years old

Heads Phoenix private-equity firm Najafi Cos. and is the vice chairman of the Phoenix Suns basketball team.

His firm says it is willing to invest up to $1 billion in companies.

Acquired Network Solutions for $20 million plus debt in 2003; sold it for $800 million in 2007.

Bought Book-of-the-Month Club, Columbia House and Doubleday Book Club from Bertelsmann AG in 2008.

Write to Mike Spector at mike.spector@wsj.com and Jeffrey A. Trachtenberg at jeffrey.trachtenberg@wsj.com