Senator Manchin is due at New Hampshire next week as the non-partisan group No Labels claims to have a path to victory should President Trump and President Biden be the nominees.

Mr. Manchin is set to make an appearance at the New Hampshire Institute of Politics’s “Politics and Eggs” breakfast next week, a venue where both Vice President Pence and attorney Robert Kennedy Jr. teased their presidential runs ahead of an announcement.

A potential presidential candidacy from Mr. Manchin has been the topic of speculation for months, and Mr. Manchin’s decision to not run for re-election and instead travel the country to “see if there is an interest in creating a movement to mobilize the middle” only added to that speculation.

Since his announcement that he would not seek re-election, however, Mr. Manchin has been mum on the prospects of a third-party bid with No Labels’s backing.

The group and its surrogates, however, have not been signaling that they intend to put up a candidate with the group claiming to have a path to victory in the event of a rematch between Mr. Biden and Mr. Trump in 2024.

 No Labels claims that it can win the plurality of the vote in 2024 with a “unity ticket,” which they project could win 34 percent of the vote to Mr. Biden’s 33.3 percent and Mr. Trump’s 32.7 percent.

The group’s model does not, however, name a specific candidate who would head up this so-called “unity ticket,” a factor that would no doubt change the support they would garner. No Labels did not immediately respond to a request for comment from the Sun.

Other surveys, such as a USA Today and Suffolk University poll conducted in late December, have shown that though there is an opening for third-party candidates, No Labels’s projection is likely optimistic.

The Suffolk University poll found that, in a seven-way matchup between Mr. Biden, Mr. Trump, the Green Party’s Jill Stein, the Libertarian Party’s Lars Mapstead, Mr. Kennedy, professor Cornel West, and an unnamed No Labels candidate, No Labels would only draw 1.6 percent support.

At the same time, a top representative for No Labels, Senator Lieberman, is insisting that the prospects for a third-party bid only increase as Mr. Biden and Mr. Trump look more and more likely to be the nominees for their parties.

On Sunday, Mr. Lieberman told radio host John Catsimatidis that he had “never seen this much anger at the two major parties, and this high disappointment that we may be left with a choice of Trump and Biden once again.”

“We want to get on the ballot so we can have the option next year … of doing something really unprecedented in America,” Mr. Lieberman said. “We’re making good progress.”

As No Labels works on recruiting a candidate that can help them realize those hopes, Mr. Kennedy is pushing to gain ballot access in all 50 states.

Wednesday, Mr. Kennedy will hold a rally in Utah, the first state where his campaign has submitted signatures to gain ballot access.

Mr. Kennedy’s campaign had sued to extend the deadline for ballot access to March 5 from January 8. After the suit, the state’s lieutenant governor, Deidre Henderson, agreed to move the deadline.

The same USA Today and Suffolk University survey found that Mr. Kennedy enjoys 10 percent support in a seven-way matchup, demonstrating the importance of the specific candidate. Mr. Trump and Mr. Biden polled at 37 percent support and 34 percent support each in the same survey.