(FORTUNE Magazine) – If the big investors who own Burger King have it their way, the next CEO of the floundering fast-food chain will be Greg Brenneman, former president of Continental Airlines. The man chasing Brenneman, David Bonderman, who heads the private-equity firm Texas Pacific Group, has required no outside headhunting help; the two men grew close a decade ago when Brenneman helped Gordon Bethune pull Continental from near bankruptcy--and made Bonderman a lot of money. Okay, but what's an airline guy going to do to revive a fast-food company? Fill seats! As McDonald's has rebounded, Burger King has been closing stores and has fallen behind Wendy's to No. 3 in annual U.S. systemwide sales. In early July majority owner Texas Pacific, with Bain Capital and Goldman Sachs Capital Partners, eased out Brad Blum, Burger King's eighth CEO in 15 years.

Brenneman, who first learned the turnaround game as a Bain consultant, has not yet signed on to fight the burger wars. But Bonderman appears to be flexing all the dealmaking muscle needed to get him. Having left Continental at the ripe age of 39 three years ago, Brenneman has played hard to get, typically bargaining for rich pay packages. A few years ago when Home Depot's directors were stressing over then-CEO Arthur Blank's weak performance, Brenneman showed no interest in coming in as Blank's No. 2. (Brenneman, a Home Depot director himself, subsequently helped push Blank out of the company.) In 2002, Brenneman briefly served as CEO of PricewaterhouseCoopers Consulting--within a month of joining, he negotiated a $3.5 billion sale to IBM.

Not to be pinned down, he has mainly stayed below the radar near Houston, investing in troubled companies through his own private-equity outfit, TurnWorks. He recently came close to taking another big CEO job, at Starwood Hotels. But when Barry Sternlicht, who is stepping down as Starwood's chief while remaining chairman, and the board dragged negotiations on for some six months, Brenneman lost patience and bowed out. Given his rallying cry at Continental--"Do it fast ... do it now!"--he will no doubt hit the ground running in fast-food land. --Patricia Sellers