Senator Manchin and a bipartisan group of national political figures will travel to New Hampshire next week to launch the political platform of No Labels, a group that is preparing to launch a third-party presidential challenge if Democrats and Republicans renominate presidents Biden and Trump.

A majority of Americans don’t want another Trump-Biden matchup, which polling suggests is increasingly likely. Democrats, though, are fearful that a No Labels fusion ticket that includes a moderate Democrat and a moderate Republican in the president and vice president slots would be a greater threat to Mr. Biden’s re-election than Mr. Trump’s.

“It will siphon off enough votes to kill somebody, could be they either murder Biden or they murder Trump,” a Democratic strategist, Hank Sheinkopf, tells the Sun. Yet he says a moderate fusion candidate is more likely to hurt Mr. Biden, because Mr. Trump’s base is “extraordinarily loyal. They represent a very strong ideological bent within the Republican Party.”

Should Mr. Manchin choose to run, the Democrat faces a tough re-election campaign in deep-red West Virginia against the state’s popular Republican governor, Jim Justice. Mr. Manchin, who boasts that he wins any election he runs in, is said to be toying with the idea of a third-party run for president.

No Labels says it is not committed to running a third-party challenge, but it is keeping the door open as an “insurance policy” in the event of a Trump-Biden rematch. “If it’s not Trump and Biden, or if Biden has a statistical lead, then we’ll throw the cards in,” a former Michigan Republican congressman and No Labels supporter, Fred Upton, told CBS.

The event Monday at St. Anselm College at Manchester will feature, among others: Mr. Manchin; a former Republican governor of Utah who was a 2012 presidential candidate, Jon Huntsman; as well as a former Democratic presidential candidate, Joseph Lieberman. The group will be announcing its “Common Sense” agenda and showcasing the potential of a bipartisan third-party ticket.

“We poll a lot that people say that they’re not really interested in a re-do of 2020,” the director of the New Hampshire Institute of Politics at St. Anselm, Neil Levesque, who is hosting the event, tells the Sun. “If Democrats are worried about something it’s because they’re not putting up a candidate that they feel confidence in.”

Democrats are indeed worried. Nearly 70 percent of Americans are concerned that Mr. Biden doesn’t have the “necessary mental and physical health to be president,” according to an NBC poll in June. Calls for Mr. Biden to drop out of the race are getting louder.

A left wing think tank, Data for Progress, released a poll in late June that shows a moderate third-party challenge could garner as much as 13 percent of the vote, siphoning off a third of independent voters and handing the election to Mr. Trump by a slim margin. The test candidate they used was a moderate former Republican governor of Maryland, Larry Hogan, who is a No Labels national co-chairman.

“It’s evident that even under the best of conditions, a moderate third-party candidate is highly unlikely to secure the vote share needed to win the White House, further underlying the nonviability of a No Label candidacy,” Data for Progress wrote. “With no feasible path to victory, such a campaign would only serve to split Independent voters, undermine Biden’s reelection campaign, and likely spoil the election in favor of Trump.”

Mr. Sheinkopf agrees, saying a third-party candidate has “none whatsoever” possibility of winning, “because voting is habitual and ritualistic.” While nearly 60 percent of Americans consistently say they’d like a third party option — and 49 percent say they’d consider voting third party in another Trump-Biden matchup — when Americans enter the voting booth their actions speak louder than words. Third parties don’t win.

No Labels is “about pressuring the parties not to renominate Biden and Trump, which is about as good as banging your head against the subway car when it’s in motion to have it stop,” Mr. Sheinkopf says. “We’ve never had, in recent memory, a successful third-party candidate run for national office.”

The chairman of the New Hampshire Democratic Party agrees, telling the Sun that the No Labels event is “a bad publicity stunt.”

“Third-party access to the ballot in New Hampshire is a year away,” Raymond Buckley says. “It doesn’t make any practical sense for them to be here, other than to try to generate attention”

Attention, though, is integral to No Labels’s mission to offer Americans an alternative to Messrs. Biden and Trump. The group has amassed a $70 million war chest and is working to get ballot access in all 50 states. So far, No Labels has succeeded in Colorado, Arizona, Alaska, Oregon, and Utah. It plans to hold a national convention in April 2024 at Dallas.

Mr. Manchin told the Washington Post earlier this year that he would not rule out running on the No Labels ticket. That he is headlining the event next week is fueling speculation he plans to run. Democrats are also worried this will lose them their majority in the Senate, as West Virginia is unlikely to elect another Democratic senator, and if Republicans were to win the White House they would get the tie-breaker vote.

A Manchin candidacy “presents some problems for Biden, because one of Biden’s strengths is appealing to somewhat conservative, moderate working-class people,” a political science professor at the University of New Hampshire, Dante Scala, tells the Sun. He also says a No Labels candidate will likely hurt Mr. Biden more than Mr. Trump, because, “Biden would want to be able to count on voters who say ‘Never Trump again,’ and would reluctantly vote for Biden despite their concerns, despite their lack of enthusiasm.”

Other potential No Labels candidates floated are Messrs. Huntsman and Hogan and even New Hampshire’s moderate Republican governor, Chris Sununu. Mr. Sununu, though, told New Hampshire Journal last week that a No Labels run is “nothing I’m considering.”

Mr. Trump failed to get 50 percent of the vote in either 2016 or 2020, so a third-party challenger could help him. It could also hurt him if the No Labels candidate were a strong Republican who could peel off some voters who aren’t thrilled with Mr. Trump but don’t want to cast their ballots for Mr. Biden.

Democrats should be concerned about a No Labels run if the group achieves its ballot access goals. Mr. Biden won the battleground states of Arizona, Georgia, and Wisconsin by less than 1 percent of the vote in 2020. He is already facing a challenge to the left by Green Party candidate, Cornel West.

Democrats are taking the No Labels threat seriously. Officials from a progressive group, MoveOn, and a moderate think tank, Third Way, are planning to brief Senate Democratic chiefs of staff on July 27 about the threat of a No Labels run, according to Politico. A former Democratic leader in the House, Richard Gephardt, is also launching a group to oppose No Labels’s work, according to the Washington Post.

“Remember when they told you a Republican ‘red wave’ was coming in the 2022 midterms? Or that Hillary Clinton had a ‘100 percent chance of winning the election’ in 2016 against Donald Trump?” No Labels’s website states. “Keep this in mind next time you hear that an independent ticket could never win the White House in 2024.”  

“The glue of partisanship right now is so strong it’s hard to imagine someone being able to put together a campaign that would peel off that many voters,” Mr. Scala says. He calls Messrs. Manchin, Hogan, and Huntsman part of the “Wayback Machine” and says it would take a “generational difference” candidate who truly inspires to actually “shake things up.”