Female Republican lawmakers are making the case for GOP presidential front-runner Donald Trump Donald John TrumpObama calls on Senate not to fill Ginsburg's vacancy until after election Planned Parenthood: 'The fate of our rights' depends on Ginsburg replacement Progressive group to spend M in ad campaign on Supreme Court vacancy MORE to choose a woman as his running mate.

Picking a female vice presidential candidate could help mitigate the damage Trump caused with women voters in the primary, they say, as well as counter Hillary Clinton Hillary Diane Rodham ClintonWhat Senate Republicans have said about election-year Supreme Court vacancies Bipartisan praise pours in after Ginsburg's death Trump carries on with rally, unaware of Ginsburg's death MORE if she is the Democratic nominee.

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Rep. Cynthia Lummis Cynthia Marie LummisChamber of Commerce endorses McSally for reelection Cynthia Lummis wins GOP Senate primary in Wyoming Chamber of Commerce endorses Ernst for reelection MORE (R-Wyo.) offered the idea, somewhat unprompted, at a Capitol Hill event hosted by the Heritage Foundation this week.

When a reporter asked a panel of conservative House Republicans who Trump should choose as his running mate, most lawmakers demurred. But Lummis, the only woman on the dais, didn’t hold back.

“I would suggest a woman, because some of the remarks he has made about women are not going to help him appeal to some of the 53 percent of the voting populace that are women,” Lummis said.

Lummis said Trump would be well served by a running mate with more “discipline” to balance his unpredictability. That would be a better strategy than going the traditional route of choosing a running mate who represents a swing state, she said.

“He has succeeded on the strength of his personality, so having a regional balance, as frequently happens, is not as important as perhaps having a personality balance,” she said.

The pressure on Trump to pick a woman has been amplified by fellow Republican candidate Ted Cruz Rafael (Ted) Edward CruzSenate Republicans face tough decision on replacing Ginsburg Cruz: Trump should nominate a Supreme Court justice next week Renewed focus on Trump's Supreme Court list after Ginsburg's death MORE, who this week named Carly Fiorina as his running mate in a bid to stop the businessman’s march to the nomination. Fiorina was the only Republican woman who ran for president this election cycle.

Recent polling shows that Trump is deeply unpopular among women, who typically vote in larger numbers than men.

A USA Today-Suffolk University poll released this week found that 66 percent of likely female voters hold unfavorable views of Trump, compared with just 24 percent who see him favorably. The same poll found that only 42 percent of women viewed Clinton favorably, but in a Clinton-Trump matchup they went for the former Secretary of State by 21 points. Clinton’s commanding lead among women contributed to her beating Trump overall, 50 percent to 39 percent.

Republican women say Trump has a long way to go in improving his standing with female voters.

“He’s said some things that have been very offensive in the past. I don’t know how you build those bridges back. Potentially he could. But it’s a little tough,” said Rep. Kristi Noem (R-S.D.), who had endorsed Sen. Marco Rubio Marco Antonio RubioFlorida senators pushing to keep Daylight Savings Time during pandemic Hillicon Valley: DOJ indicts Chinese, Malaysian hackers accused of targeting over 100 organizations | GOP senators raise concerns over Oracle-TikTok deal | QAnon awareness jumps in new poll Intelligence chief says Congress will get some in-person election security briefings MORE (R-Fla.) before he ended his campaign in March.

Still, Noem said that Trump would send a powerful message by choosing a woman as his vice president.

“I just feel like whenever you have a woman at the table, you have a better perspective on every single issue. Women are the majority of the voters in this country. Whenever I’ve been in a discussion where women and men are at the table, I feel like it’s more rounded. It’s more collaborative,” Noem said. “So I think it would bring a lot to his ticket if he were able to do that.”

There are a number of prominent Republican women who could land on Trump’s short list. Lummis floated former Secretary of State Condoleezza Rice and Sen. Kelly Ayotte Kelly Ann AyotteBottom line Bottom line Bottom Line MORE (R-N.H.) as possibilities.

Ayotte is considered one of the most vulnerable Senate Republicans up for reelection in November, but it’s unclear whether she’d be open to running with Trump. She has not endorsed any candidate in the GOP primary but has said she will support the eventual nominee.

“Kelly is focused on doing her job serving the people of New Hampshire in the Senate, and she is running for reelection,” Ayotte campaign spokeswoman Liz Johnson said in response to the vice presidential idea.

Rice, meanwhile, has shown little interest in elected office since her days in the George W. Bush administration. And Trump could have trouble convincing her to join his ticket, given his criticism of the Bush family, which Rice served in the White House.

House Administration Committee Chairwoman Candice Miller (R-Mich.), who is retiring at the end of this year, suggested New Mexico Gov. Susana Martinez as a possible pick. However, Martinez had endorsed Rubio before he dropped out of the race, and she has sharply criticized Trump for his remarks on immigration.

“It’s his personal decision to make,” Miller, who had endorsed Fiorina, said of Trump choosing a female running mate. “But I do think we have qualified women. And if he chose one, I would be delighted.”

Sen. Susan Collins Susan Margaret CollinsSenate Republicans face tough decision on replacing Ginsburg Democratic senator calls for eliminating filibuster, expanding Supreme Court if GOP fills vacancy What Senate Republicans have said about election-year Supreme Court vacancies MORE (R-Maine) emphasized that Trump should choose someone with governing experience regardless of gender.

“I think Donald Trump is likely to be the nominee and that he should choose someone who has experience governing, either at the state level such as a governor or here in Washington, to help balance out his experience,” Collins said.

Yet Collins added: “For some people, I’m sure it would be exciting to see a woman in the vice presidential slot on the Republican side.”

Trump has said that he would prefer a running mate with more political experience. Trump told The Washington Post in a recent interview that he’d “95 percent see myself picking a political person as opposed to somebody from the outside.”

And in an interview with USA Today, he floated former presidential rivals Rubio and Wisconsin Gov. Scott Walker, as well as Ohio Gov. John Kasich, who’s still in the race.

Another frequently mentioned name is Rep. Marsha Blackburn Marsha BlackburnDemocrats smell blood with new DHS whistleblower complaint Hillicon Valley: Election officials prepare for new Russian interference battle | 'Markeyverse' of online fans helps take down a Kennedy | GOP senators unveil bill to update tech liability protections GOP senators unveil new bill to update tech liability protections MORE (R-Tenn.), who is serving as chairwoman of a House panel investigating abortion service providers.

When asked in a brief interview about Trump choosing a female running mate, Blackburn expressed support for the idea.“Absolutely,” she said.

South Carolina Gov. Nikki Haley is another oft-mentioned possibility, but she delivered a stinging rebuke of Trump in the GOP response to President Obama’s State of the Union address, warning the party should not embrace its “angriest voices.”

Trump has expressed openness to a female running mate. The candidate himself even mentioned one idea early in his presidential campaign: Oprah Winfrey. But given the TV star’s lack of political experience and previous support for Obama, she might not be on Trump’s list anymore.

“I think Oprah would be great. I'd love to have Oprah," Trump told ABC News’ George Stephanopoulos in June. “I think we'd win easily, actually.”