Opinion is still split on whether Ms. Fiorina or her successor as chief executive, Mark V. Hurd, deserve credit for Hewlett’s success after Ms. Fiorina drove through the company’s $25 billion acquisition of Compaq in 2002. By many accounts, Ms. Fiorina was superb at marketing, mixed on strategy, bad at execution  and extraordinarily successful in unifying the board against what Jeffrey A. Sonnenfeld of the Yale School of Management calls her “street bully” leadership style.

Image Carly Fiorina, an adviser to Republican presidential candidate Senator John McCain, on Mr. McCain's campaign bus. Credit... Mary Altaffer/Associated Press

“What a blind spot this is in the McCain campaign to have elevated her stature and centrality in this way,” said Mr. Sonnenfeld, the senior associate dean for executive programs at the management school and one of Ms. Fiorina’s sharpest critics. “You couldn’t pick a worse, non-imprisoned C.E.O. to be your standard-bearer.”

But Mr. McCain, as Ms. Fiorina put it, does “clearly not” share the views of her critics. To the contrary, he so proudly calls on Ms. Fiorina in her regular appearances with him on the campaign trail  he calls her an American success story “who began as a part-time secretary”  that he seems to be suggesting that Ms. Fiorina, true or not, might have a role in a McCain cabinet.

As a result, Ms. Fiorina has been buzzed about as a potential commerce or Treasury secretary or even as a McCain running mate, although some Republicans close to Mr. McCain swiftly dismiss the idea of her as vice president. But the view within the campaign is that it can only help Mr. McCain’s standing among women to have Ms. Fiorina mentioned as a possibility for high-profile office in a McCain administration, particularly when he is trying to win over the supporters of Senator Hillary Rodham Clinton.

In the meantime, Ms. Fiorina has done little to tamp down speculation that she might run for office herself, including the California governorship in 2010.

“I would be disingenuous if I said it has never occurred to me,” Ms. Fiorina said about running in general. “And in part it occurs to me because people keep asking. When I give speeches, people raise their hand  ‘run, run, run.’ ” For now, she said, “I’m focused on getting McCain elected.”