Some prominent Republicans said it would be helpful for the party to have a woman running for president, especially considering the expected candidacy of Hillary Rodham Clinton on the Democratic side. But they questioned whether Fiorina is the right woman.

At Hewlett-Packard, Fiorina was a pioneering executive — the first female CEO of a Fortune 50 company — but her high-profile tenure was controversial. In 2005, after a merger with Compaq, she was forced to resign.

After serving as a prominent surrogate for Sen. John McCain (R-Ariz.) in his 2008 presidential campaign, Fiorina made her first run for elected office in 2010, challenging Sen. Barbara Boxer (D-Calif.). She staked out conservative positions to the right of California’s mainstream — opposing abortion rights and efforts to cut greenhouse gases, for example — and lost to Boxer by 10 points.

Reed Galen, a California-based Republican strategist, said Fiorina is “obviously very interesting, very dynamic and, as one of the first female CEOs, has a good story to tell.” Asked to describe her base within the GOP primary electorate, Galen said: “I’m not sure. My inability to answer shows you how hard a road she has.”