The attention-seeking West Virginia senator, who has teased a late third-party presidential bid, tried to keep up the suspense at a Friday appearance in the state.During an eyebrow-raising visit to New Hampshire on Friday, Senator Joe Manchin III of West Virginia name-checked friends who are elected officials in the Granite State and complimented the discerning nature of its voters.He paid homage to the state’s first-in-the-nation primary tradition and swiped at President Biden’s decision to undercut New Hampshire’s power in this year’s Democratic contest.And when pressed on his own ambitions, the conservative Democratic senator offered a message that would-be candidates have often deployed as they flirt with this historically influential early-voting state: He declined to rule anything out.“How would you feel if a bunch of Democrats in New Hampshire wrote in ‘Joe’ — not Biden — but wrote in ‘Joe Manchin’?” an attendee asked as Mr. Manchin kicked off a “listening tour” at Politics and Eggs, an event series at the New Hampshire Institute of Politics that has long hosted presidential candidates and potential contenders.“I cannot prevent whatever you want to do,” Mr. Manchin replied to applause from the audience in Manchester, N.H., before insisting that he was “not here campaigning.”The question of what Mr. Manchin wants to do has long infuriated and confounded many of his Democratic colleagues in Washington, who have often seen him as a roadblock to their legislative agenda, even as he has played a pivotal role in eventually passing key priorities.Now, Mr. Manchin — known for a love of the spotlight that stands out even among U.S. senators — is stoking new questions about his next steps.Speculation has grown about whether he might embark on a late, long-shot presidential bid this year, and he has attracted interest from No Labels, a centrist group that is searching for a “unity ticket” to mount a potential third-party bid. Democratic allies of Mr. Biden are trying to stave off such efforts.Mr. Manchin, who announced in November that he would not seek re-election in his deep-red state in 2024, has teased a potential third-party run for the presidency.Charles Krupa/Associated Press“He really deserves most serious consideration from No Labels because he is part of our movement” if he is interested in a third-party bid, said former Senator Joseph I. Lieberman of Connecticut, the founding chairman of the group. He said he had spoken with Mr. Manchin after the senator announced in November that he would not seek re-election. “He’s walked the centrist, bipartisan, problem-solving walk.”(Mr. Lieberman has also talked up a run by former Gov. Chris Christie of New Jersey, who dropped out of the Republican presidential race this week. But he said on Thursday that while Mr. Christie had many fans at No Labels, the last time he had personally spoken to him was probably “at a Mets game last summer.”)Mr. Manchin did not offer a ringing endorsement of the group’s plans when asked on Friday about the electoral potential of such a bid.“It’s admirable what they’re trying to do to provide an option — OK, they’re working very hard towards that, and their best intentions are to bring people together,” he said, noting his longtime involvement with the group. Pressed again on the question of viability, he replied: “I don’t know. I mean, you have to — the people decide that. I think by Super Tuesday, you’ll know what’s going on.”Mr. Manchin, with his daughter, has started an organization called Americans Together, designed to elevate moderate voices — the “responsible, sensible, common-sense middle,” he said on Friday — whom he casts as often politically homeless. The New Hampshire swing was the first stop on what his team has called a listening tour, but he emphasized that his group was “completely different” from No Labels.Throughout his appearances — at the breakfast, in speaking with reporters and at a diner where he was trailed by climate-focused protesters — Mr. Manchin denounced the far right and the far left (though any notion that Mr. Biden falls close to that category is risible to his many left-wing detractors). And at times, Mr. Manchin seemed to slip into overt campaign mode, even as he insisted at other points that he had made no decisions about a run.“Everyone says, ‘Well, are you running for this, or running for that?’” he said on Friday morning, adding that, no, he was “running” to “bring the country together.”“I want you to know there’s hope,” he continued. “Nobody can win up here unless they get the independent vote. Nobody can win unless they get the center left and center right.”He also repeatedly declined to say whether he would support Mr. Biden over former President Donald J. Trump in a November matchup, though he has said in the past that he will not back Mr. Trump.“I’m not picking anything right now until we see what we have,” he said, though he later allowed in an interview that he was “absolutely comfortable with Biden’s character.” He added: “Do I agree with the politics? Not all of the time.”He also nodded to recent polls that have shown Mr. Biden struggling, calling them “alarming,” adding, “The whole thing is alarming, from a standpoint, how close it can be again, how it might even flip to a different direction.”Democrats worry that third-party bids could siphon votes from Mr. Biden and hand the election to Mr. Trump if he is the Republican nominee. Matt Bennett, a founder of the center-left group Third Way, who has been engaged in efforts to block third-party and independent candidates, expressed optimism that Mr. Manchin would not go that route. Mr. Manchin, for his part, has insisted that he has no interest in being a “spoiler.”“Joe Manchin is on a listening tour to talk to voters about the value of moderate ideas, and we think that’s fantastic,” Mr. Bennett said in a text message. “We think it’s smart for him to have started in N.H. and get the attention from the giant political press corps there. We know he hasn’t made a final decision on running for president, but we’re confident that he won’t.”Mr. Manchin suggested on Friday that the country was interested in more options, but he seemed uncomfortable directly engaging in talk of a third-party bid himself, saying vaguely at one point: “There might be more choices. There might be different choices. We just don’t know yet.”In an interview, he said: “I’m looking for, how do you bring the country together, how do we get people involved? And if that’s a decision to make, I’ll live with whatever decision.”As he wrapped up glad-handing at the diner in Derry, where he told a Republican fan that he did not know if he would run, a reporter asked if he could name one thing that appealed to him about a third-party bid and one thing that would give him pause.The usually voluble senator smiled, declared that he was there to bring Americans together and walked away. More