President Biden is making overtures to Governor Haley supporters since she dropped out of the presidential race Wednesday morning — but the addition of a No Labels moderate Republican presidential candidate could stymie those efforts.

Democrats are worried. No Labels’ 800 delegates will meet virtually on Friday to vote on whether to run a third-party “unity ticket,” with a Republican and Democrat in the presidential and vice-presidential slots. A former Democratic mayor of Dallas, Michael Rawlings, will moderate the event.

The nonprofit group — it is not a political party and therefore don’t have to disclose donors — has for months teased an announcement after Super Tuesday, but it is now clear that No Labels itself isn’t sure whether it will run a presidential ticket. No Labels chief strategist, Ryan Clancy, called Friday’s meeting a “gut check” on C-Span Wednesday morning.

“This meeting is a chance for our delegates to speak freely and honestly about the path ahead for our 2024 project,” Mr. Clancy tells the Sun. “Candidates will not be chosen for the Unity ticket during this meeting and it will not be open to the press.”

No Labels will not disclose who its delegates are or which candidates it is considering. A former Maryland governor, Larry Hogan, who is a moderate Republican, stepped down from a leadership position in No Labels in December and announced he is running for the United States Senate. The moderate Democratic senator, Joe Manchin, also declined to run on a No Labels ticket.  

No Labels has also floated Mrs. Haley, but she says she is not interested. No Labels posted to X Wednesday to congratulate Mrs. Haley “on running a great campaign and appealing to the large swath of commonsense voters.”

“We take her at her word that she isn’t interested in pursuing another route to the presidency,” No Labels posted. “We will have more to say when the meeting concludes on Friday.”

Democratic strategist Hank Sheinkopf tells the Sun he thinks No Labels will pull the trigger and run a presidential ticket. “They’re going to do whatever they can to keep their label alive,” he says. “Not doing it would be admitting defeat. It would also be telling their funders that they wasted their money, and those heavily engaged would never do that.”

“At best they will injure Joe Biden. At worst, they will help elect Donald Trump,” Mr. Sheinkopf says.

In a sign of how worried Democrats are, a group of mainly moderate Democrats, led by a former Democratic congressman, Dick Gephardt, sent an open letter to No Labels on Tuesday asking the group’s leadership to sign a “No Spoiler Pledge” at its virtual convention Friday.

“By voting to sign the No Spoiler Pledge, you will commit No Labels to not be a spoiler in this pivotal election for our democracy by returning Donald Trump to the White House,” the letter says. “If by July 1, No Labels and/or its ticket has not qualified to be on the ballots in enough states to have a mathematical chance to win 270 electoral votes and is not competitive in five or more states, No Labels commits to withdrawing its candidate from the ballots in the swing states that will decide the election.”

No Labels did not return the Sun’s request for comment on the letter. 

Mr. Clancy has repeatedly said that 2024 is a unique year in that a majority of Americans do not want another Trump-Biden matchup. He rejects the “spoiler” narrative, saying that No Labels will only run a ticket with a “viable path” to victory. Proponents of No Labels cite Gallup polling that shows 63 percent of Americans think a third party is needed, and a record high of 43 percent of Americans now identify as independents.  

The group has already secured ballot access in 16 states and is working on getting it in 17 others. In several states, a candidate must be named before attaining ballot access.

 “We’ve said all along that we will only put up a ticket if we think there’s an opening. If it’s Trump and Biden, we think there is an opening, at least in the numbers,” he told the Sun in January.

Mr. Sheinkopf says that’s foolhardy. “This is an intellectual exercise in what is an emotional process. People vote from their gut, not from their brain,” he says. “We haven’t had a third party that functioned and took control of any other party since the destruction of the Whigs and the creation of the Democratic Party by Martin Van Buren. I mean, Teddy Roosevelt tried it. It didn’t work.”