Investors have been speculating for several months about whether Gap might oust Pressler or consider takeover offers in response to the clothing retailer's second prolonged sales slump in the past six years.
The latest funk began in the spring of 2004, prompting repeated promises of a turnaround.
The troubles instead have been deepening, culminating in a miserable holiday shopping season that forced Pressler last week to lower the company's earnings guidance for the third time in the last five months.
Gap's profit for its fiscal year ending in January is expected to fall about $300 million, or 40 cents a share, below what Pressler envisioned when 2006 began.
A number of candidates are being discussed for the top job at Gap, according to CNBC's Margaret Brennan. She cited Ralph Lauren Chief Operating Officer Roger Farah and Payless Shoesource Chief Executive Matt Rubel as examples. Allen Questrom, who is known for turning around retailers such as Barney's and Federated, and Vanessa Castagna, who is about to leave Mervyn's after orchestrating a turnaround there, are also potential candidates, she said.
Paul Charron, a former Liz Claiborne CEO, also has been widely seen as a potential candidate.
"Gap is a great property in troublesome circumstances," Charron said in an email to Brennan. "Whoever is tasked with leading this enterprise will have a number of considerable advantages, but some daunting challenges. New strategies are clearly required."
"For me, I am going to maintain a low profile and try to figure out how best to spend my time post-Liz," Charron said.
Some industry analysts have theorized that a deep-pocketed private equity firm might be willing to buy Gap for about $20 billion, or $24 to $25 a share, and then bring in new management to tackle the problems that Pressler and his team so far have unable to address.
Despite its recent woes, Gap remains an appealing takeover target because it generates lots of cash and retains a well-known brand backed by a retailing network spanning nearly 3,200 stores.
But other retail industry analysts expect Gap is becoming an increasingly unlikely leveraged buyout candidate.
"It's a lot easier to attract talent by saying come in and turnaround a business where the margins are eroded," said Gabrielle Kivitz, an analyst at Deutsche Bank, in an interview on CNBC
Besides the Gap chain, the company also owns Old Navy, Banana Republic and Forth & Towne.