If Nikki Haley doesn’t win the GOP presidential nomination — which current polls indicate is likely — would she run on a bipartisan No Labels ticket?

The former senator and co-chairman of No Labels, Joe Lieberman, floated that possibility Monday on Fox News’s “Your World” with Neil Cavuto. While he said he hasn’t spoken directly with the former South Carolina governor about it, Mr. Lieberman said he wouldn’t be surprised if some of his third-party’s financial backers have.

“I have a lot of admiration for Nikki Haley. I haven’t talked to her at all about this, but it is true that some of our big supporters know her well,” Mr. Lieberman said. “I wouldn’t be surprised if they’re talking with her.”

Ms. Haley is rising in the polls and has become the favored candidate of big-dollar donors in recent weeks who don’t want another Trump or Biden presidency. No Labels has not yet committed to running a presidential ticket in 2024, but if early primary wins suggest the 2024 presidential race will be another Biden-Trump matchup, a No Labels ticket becomes more likely.

“She’s totally focused on the Republican primary campaign, and right now she seems to be doing pretty well,” Mr. Lieberman said. “If for some reason that didn’t work out and we decided we were going to run a ticket, I think there would be a lot of people in No Labels who would be happy to give a lot of consideration to Nikki Haley.”

The suggestion that Ms. Haley would be a top choice for a third party that bills itself as centrist and bridging the partisan divide comes at an unusual time. Americans for Prosperity, a libertarian-conservative political advocacy group founded by the Koch brothers — the boogeymen of the left — endorsed Ms. Haley last week. That endorsement would have been kryptonite to a Democrat or independent in a pre-Trump era.  

Yet anti-Trump sentiment is stronger. Polls show that a majority of Americans don’t want either President Biden or President Trump to run, and both have high unfavorables. This is the premise of which No Labels sees a potential third-party victory, citing the two incumbents’ “profound vulnerabilities that likely will not get better with time.”

Ms. Haley’s press secretary, Ken Farnaso, though, dismissed a No Labels run. “She is going to be the Republican nominee, not the No Labels nominee,” he tells the Sun.

Mr. Lieberman said No Labels is focusing now on ballot access and will convene a committee in January to start narrowing the field of potential candidates. Other names floated by No Labels are a former Maryland Republican governor, Larry Hogan; the former Utah Republican governor, Jon Huntsman; and West Virginia’s Democratic senator, Joe Manchin, who announced last month he is not running for re-election. That fueled speculation that Mr. Manchin will run at the top or bottom of a 2024 No Labels presidential ticket. The party’s nominating convention is scheduled for April.

“No Labels will have discussions with many candidates in the coming months. Over 1,000,000 voters have now signed No Labels petitions across the nation,” No Labels’s chief strategist, Ryan Clancy, tells the Sun. “We are on the ballot in 12 states, and we will be active in 27 states by the end of 2023.”

No Labels dismisses the “spoiler” narrative, saying a Trump-Biden matchup in 2024 is a unique situation in which “an independent presidential candidate has a chance to win the support of as much as three in five American voters.” A ticket with Ms. Haley at the top and a Democrat like Mr. Manchin as the vice-presidential pick may take more votes away from Mr. Trump, not Mr. Biden.

That Ms. Haley would appeal to centrist voters looking for an alternative to Messrs. Biden and Trump is not surprising. She has attempted to stake a more moderate position on abortion than the other Republicans in the field. At a recent campaign event at New Hampshire, the Sun spoke with several attendees who all said they didn’t want to vote for Mr. Trump or Mr. Biden and were looking for an alternative. Some of them were Democrats.

“I’m definitely against Trump. Didn’t vote for him, wouldn’t vote for him. I did vote for Biden but I always register as independent,” a Granite State realtor, Kathy Sutherland, told the Sun. She said she does “always end up leaning toward Democrat voting” but “watched Nikki Haley on the last debate and she intrigued me.”

“I just joined the No Labels movement. Just gathering information on that and trying to understand where they’re coming from,” another New Hampshire resident at the event, Doug Houghton, told the Sun. He said he used to be Republican but has voted Democrat “the last couple elections.”

Other attendees said they were also interested in Governor Christie and Mr. Manchin, if he runs on a No Labels ticket. These voters were not interested in Governor DeSantis or Vivek Ramaswamy.

Some Republican donors and strategists are now calling on Mr. Christie to drop out the race so the anti-Trump vote can coalesce around Ms. Haley. Whether the last-minute infusion of cash and organizational power from Americans for Prosperity can help Ms. Haley beat Mr. Trump still looks doubtful.

There is also the issue of whether Republican voters would coalesce around Ms. Haley if the other second-tier candidates dropped out of the race. The grassroots director for Americans for Prosperity’s New Hampshire chapter, Chris Maidment, posted to X late Friday his dismay at the organization’s endorsement of Ms. Haley.

“Nikki Haley is totally sideways on Americans for Prosperity’s foreign policy stance. She’s anti free speech,” Mr. Maidment posted to X, saying he’s “lost faith” in the organization. “If it comes down to it, I’ll vote for Biden over Haley 10 times out of 10, because he’s less of a threat to our country.”

“Americans for Prosperity has always pushed forward foreign policy that is realistic and restrained. And she cuts through that grain in a large way,” Mr. Maidment tells the Sun. “I know several folks that disagree with the endorsement but won’t say it because they believe in the organization.”

A former director of the Iowa chapter of Americans for Prosperity, Mark Lucas, endorsed Mr. Trump at an event in Iowa Saturday. “Respectfully, anybody not supporting Trump is delaying the inevitable,” Mr. Lucas said.