Almost since he arrived in Washington, Senator Joe Manchin III of West Virginia has complained about the partisan nature of the Capitol and insisted that Americans aren’t as politically divided as the people they send to Congress.

With his announcement on Thursday that he will not seek re-election next year, Mr. Manchin again floated the possibility that he thinks the solution to America’s polarized politics lies in the mirror.

“What I will be doing is traveling the country and speaking out to see if there is an interest in creating a movement to mobilize the middle and bring Americans together,” Mr. Manchin said in his retirement video.

He added, “I know our country isn’t as divided as Washington wants us to believe. We share common values of family, freedom, democracy, dignity and a belief that together we can overcome any challenge. We need to take back America and not let this divisive hatred further pull us apart.”

What Mr. Manchin actually plans to do remains a mystery. His closest aides and advisers insist they don’t know. A conservative Democrat who has served as one of his party’s key votes in the Senate, he has long kept his own counsel on his biggest decisions and made up his mind at the last minute.

Mr. Manchin has flirted this year with No Labels, a group that has made noise about running a centrist candidate for the White House. No Labels officials said Thursday that Mr. Manchin’s announcement had taken them by surprise, though they commended him “for stepping up to lead a long-overdue national conversation about solving America’s biggest challenges.”

“Regarding our No Labels Unity presidential ticket, we are gathering input from our members across the country to understand the kind of leaders they would like to see in the White House,” the group said in a statement.

Some allies of Mr. Manchin are skeptical that he will run for president. For one, it would cost hundreds of millions of dollars to run a credible independent or third-party campaign, and Mr. Manchin has never been a formidable fund-raiser on his own.

Fellow Senate Democrats and their super PAC subsidized much of his 2018 re-election effort and were poised to do so again next year had he chosen to run. He did hold a fund-raising event for his political action committee last weekend at the Greenbrier, the West Virginia resort owned by Gov. Jim Justice, a Republican who is running for the state’s Senate seat.

But the odds of him winning the presidency would be extremely long, especially at this relatively late date.

“I wouldn’t say that he can’t or won’t run, but I know he hasn’t run for anything that he doesn’t want to win, ever,” said Phil Smith, a longtime lobbyist and official at the United Mine Workers of America and an ally of Mr. Manchin’s. “If you look at independent candidates for president, even well-known ones, those who started this late never got more than 2 to 3 percent of the vote.”

Then there’s the question of Mr. Manchin’s age. He is 76, and would be running in a race with heightened attention and concern about the ages of President Biden, 80, and the likely Republican nominee, former President Donald J. Trump, 77.

Mr. Manchin, a former West Virginia University quarterback, remains in good physical condition for a septuagenarian. In May, he completed a three-mile race in Washington in just over 40 minutes.

One thing Mr. Manchin has always enjoyed since he won a special election to the Senate in 2010, when he was West Virginia’s governor, is the attention that comes with being a critical vote when Democrats control the chamber.

That has often afforded him a platform that has made him popular among cable television bookers and centrist donors, while drawing the ire of the Democratic Party’s progressive activists. He said this summer that he was thinking “seriously” about leaving the Democratic Party.

“If he sees that Biden continues to be the Democratic nominee and Trump the Republican nominee, I think he truly sees a huge slice of the American electorate, both Republican and Democratic, fed up with both of their parties’ nominees,” said former Representative Nick Rahall, a fellow West Virginia Democrat who has known Mr. Manchin for decades.

For months this year, Mr. Manchin has cozied up to No Labels, which has so far secured ballot access in 12 states in its attempt to offer an alternative to Mr. Biden and Mr. Trump. The group’s president, Nancy Jacobson, has told potential donors that the group intends to select a Republican to lead its ticket, a decision that would exclude Mr. Manchin if No Labels maintains that position.

One candidate openly teasing a No Labels run, Larry Hogan, the former Republican governor of Maryland, released a foreign policy video on Tuesday that looked and sounded like a campaign advertisement, denouncing the isolationism in his party and declaring himself “a Reagan guy.”

Mr. Hogan appeared at a Bloomberg event last month and said that when he spoke with No Labels officials and donors, “most of them are now assuming it should be a Republican at the top of the ticket.”

No Labels has methodically moved forward on its possible presidential campaign, unveiling a manifesto — a platform of sorts — in July and holding its own centrist events. They have featured a rotating cast of characters including Mr. Manchin, Mr. Hogan and Jon Huntsman Jr., a former Utah governor and moderate Republican.

The group plans to raise $70 million before a convention in Dallas scheduled for April. But No Labels officials say they will decide whether to announce that campaign before then, possibly after Super Tuesday on March 5, when the Republican presidential primary contest may be all but over.

The decision could come earlier, with the field of presidential candidates outside the major parties continuing to expand.

On Thursday, Jill Stein, whose presence on the ballot in 2016 may have helped secure the White House for Mr. Trump, joined Robert F. Kennedy Jr., the iconoclastic vaccine skeptic, and Cornel West, the left-wing academic, as challengers to the Republican and Democratic nominees. Ms. Stein will seek to represent the Green Party, as she did in 2016.

But No Labels’s drive to get a slot on the ballot in all 50 states appears to have stalled at 12. Thirty-four states allow a group like No Labels to claim a place-holder slot without a candidate, but 16 others and the District of Columbia require a ticket.

“They’re not going to run a 50-state campaign,” said Mr. Smith, the lobbyist and union official. “They’re just not.”

There will be no shortage of unsolicited advice for Mr. Manchin from Democrats when it comes to his plans.

Matt Bennett, the co-founder of the centrist Democratic group Third Way, who is organizing efforts to stop Third Way and dissuade Mr. Manchin from joining their ticket, said he was “not worried” about Mr. Manchin running as an independent candidate.

Rahna Epting, the executive director of the progressive group MoveOn, said Thursday that Mr. Manchin should “reject any overtures from No Labels’s dangerous ploy.”

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