If Romney picks a woman, who?

Would picking a woman as his running mate help Mitt Romney close the gap with female voters?

As speculation about the No. 2 spot on the Romney ticket ramps up, the names of a handful of prominent — and some not-so-prominent — Republican women have emerged as possible vice-presidential contenders, including former Secretary of State Condoleezza Rice and two first-term governors, Susana Martinez of New Mexico and Mary Fallin of Oklahoma.

Miss Rice was the top vice-presidential pick among Republicans and independents surveyed in a CNN/ORC International poll released Wednesday, ahead of Rick Santorum, Sen. Marco Rubio of Florida and New Jersey Gov. Chris Christie.

Like almost everyone who has been mentioned as potential vice-presidential candidates, Miss Rice, now an analyst at Stanford University’s Hoover Institution, insists she is not seeking the job.

“I think we should go another direction and find somebody who really wants to be in elected office,” she said last month on Fox News. “How many ways can I say it? Not me.”

Mrs. Martinez, the nation’s first Hispanic female governor, could help Mr. Romney shore up support in two voting blocs where the former Massachusetts governor trails President Obama badly. But she said earlier this month there is no chance she will accept the slot.

Mrs. Fallin, the first female governor of Oklahoma, said Wednesday she is flattered to be mentioned as a possible running mate but she, too, downplayed the talk.

“That is all speculation at this point,” she said, but she didn’t rule out accepting an invitation to join the ticket.

“The presidential campaign is very important,” she said. “I will do everything I can to help elect the Republican nominee.”

Other Republican women who have drawn interest include South Carolina Gov. Nikki Haley, long shots such as Rep. Cathy McMorris Rodgers of Washington state and even some relative unknowns, such as Florida Attorney General Pam Bondi, an early and enthusiastic Romney backer in her state.

Rep. Rodgers, vice chairman of the House Republican Conference and the highest-ranked woman in the GOP caucus, told The Washington Times-affiliated “America’s Morning News” radio broadcast on Thursday that the vice presidency is “not on [my] radar screen.”

“I don’t expect to get that call,” the 42-year-old Spokane congresswoman told hosts John McCaslin and Dana Mills.

But she did argue that President Obama’s appeal to women voters has been overstated.

“I’ve been speaking out a lot against this phantom ’war on women’ that the Democrats suggest that the Republicans are waging. It once again is just another distraction,” she said.

She predicted that female voters will rally to Mr. Romney because of his promise to dismantle Mr. Obama’s health care law.

“Women do not like this health care bill. I really point to the health care bill as one of the reasons women voted Republican in 2010.”