NEW ORLEANS — Addressing reporters at a meeting of the Democratic Governors Association, Gov. Roy Cooper of North Carolina gave an emphatic answer when asked whether he expected President Biden to seek a second term — and whether he believed that was in the best interests of his party.

“I do and I do,” Mr. Cooper said on Friday, adding, “I fully expect him to seek re-election and I will support him, and in fact we’re going to win North Carolina for him.”

But just three minutes later, Mr. Cooper — the only Democratic governor to twice win a state that former President Donald J. Trump carried in both of his campaigns — was sketching out what could be the makings of a Cooper for President message to primary voters.

He trumpeted his repeal of his state’s so-called bathroom bill targeting transgender people, an executive order granting paid parental leave to state employees and another order putting North Carolina on a path to carbon neutrality by 2050. “That’s why Democratic governors are so important,” he said, alluding to next year’s midterm elections.

Publicly, Mr. Cooper and other Democratic leaders are focused on what will be a difficult 2022 if Mr. Biden’s popularity does not pick up. However, it is 2024 that’s increasingly on the minds of a long roster of ambitious Democrats and their advisers.

With Mr. Biden facing plunging poll numbers and turning 82 the month he’d be on the ballot, and Vice President Kamala Harris plagued by flagging poll numbers of her own, conversations about possible alternatives are beginning far earlier than is customary for a president still in the first year of his first term.

None of the prospects would dare openly indicate interest, for fear of offending both a president who, White House officials say, has made it clear to them that he plans to run for re-election and a history-making vice president who could be his heir apparent. No president since Lyndon Johnson in 1968 has opted not to run for re-election.

Still, a nexus of anxious currents in the Democratic Party has stoked speculation about a possible contested primary in two years. On top of concerns about Mr. Biden’s age and present unpopularity, there is an overarching fear among Democrats of the possibility of a Trump comeback — and a determination that the party must run a strong candidate to head it off.

Should Mr. Biden change his mind and bow out of 2024, there is no consensus among Democrats about who the best alternative might be.

The list of potential candidates starts with Ms. Harris and includes the high-profile transportation secretary, Pete Buttigieg — the two candidates most discussed in Washington — as well as a collection of former presidential candidates like Senators Elizabeth Warren of Massachusetts and Amy Klobuchar of Minnesota.

Senator Richard Blumenthal of Connecticut said that if such a race unfolded, it would be “a real mud fight in the good old-fashioned sense of Democratic fights.” If there “ever were rules” in presidential nominating contests, he added, “they no longer hold.”

Two Democrats who ran for president in the last election said they fully anticipated Mr. Biden would run again, but they notably did not rule out running themselves if he declined to do so.

“He’s running, I expect to support him and help him get re-elected,” Ms. Warren said. “I’m sticking with that story.”

Ms. Klobuchar, who told influential Democrats last year that she’d be interested in running again, said of Mr. Biden: “He has said he’s going to run again, and I take him at his word, and that’s all I’m going to say.”

A number of well-known party officials, Mr. Biden most notable among them, deferred to Hillary Clinton in 2016, leaving a sizable opening in the field that was filled by Senator Bernie Sanders. The surprising strength of Mr. Sanders’s candidacy and Mrs. Clinton’s subsequent loss to Mr. Trump upended assumptions about what was possible in today’s politics and soured many in the party on coronations.

Similarly, the meteoric rise of Mr. Buttigieg in the 2020 primary has emboldened aspiring Democrats, who took the prominence of an under-40 mayor of a small city as yet more evidence that voters have a broad imagination about who can serve as commander in chief.

Most delicate for Democrats are Ms. Harris’s struggles and the question of whether she would be the most formidable post-Biden nominee. In a party that celebrates its diversity and relies on Black and female voters to win at every level of government, it would be difficult to challenge the first Black and first female vice president.

Yet recent history provides few examples of vice presidents who have claimed the White House without a strenuous nomination fight. The last two vice presidents to win the presidency, George H.W. Bush and Mr. Biden, faced tumultuous primary contests on their way to the White House.

There is little reason to expect a smoother path for Ms. Harris.

Even Ms. Harris’s allies are alarmed at the steady stream of stories about her difficulties and a recent staff exodus.

“Everything must change, from optics to policy to personnel,” said Donna Brazile, a former Democratic National Committee chair who is close to Ms. Harris’s advisers. “She’s done a lot of good stuff, but no one talks about the achievements.”

“If Biden announces that he will not run in 2024,” she added, “it’s open sesame.”

Potential aspirants could include other figures in the Biden administration.

Mitch Landrieu, the former mayor of New Orleans who is now leading the implementation of Mr. Biden’s trillion-dollar infrastructure law, considered running for president in 2020, and some of his allies have quietly promoted him as a potentially formidable candidate in the future.

Mr. Landrieu rebuilt his city after the ravages of Hurricane Katrina and drew national acclaim for an address he delivered in 2017 heralding the removal of Confederate statues from New Orleans.

The Rev. Al Sharpton said Mr. Landrieu would be “a very interesting candidate” if Mr. Biden did not run again.

“He knows how to work the South; he knows how to work with Black and brown communities,” Mr. Sharpton said. “And having a high-profile position on infrastructure doesn’t hurt.”

Mr. Sharpton said that he heard regularly from Ms. Harris and that Mr. Buttigieg, who struggled to win even nominal support from voters of color in 2020, “has stayed in touch on a monthly basis.”

Mr. Biden’s commerce secretary, Gina Raimondo, has also expressed interest in the White House in the past.

In the run-up to the 2020 election, Ms. Raimondo, then the governor of Rhode Island, told an informal adviser that she believed there was a path to the presidency for someone of her experience and background. But Ms. Raimondo, a leader of her party’s moderate wing, recently told an associate she was “out of the politics business.”

Yet should Mr. Biden rule out a second campaign, there are also Democrats who believe the party would be better off turning to a leader from outside Washington rather than recruiting from within a weary administration.

At the governor’s conference in New Orleans over the weekend, circumspect questions about Mr. Biden’s age and Ms. Harris’s vulnerabilities dotted the corridor and cocktail conversations.

Mr. Cooper already has donors encouraging him to consider a bid, according to Democrats familiar with the conversations.

Should Gov. Gretchen Whitmer of Michigan survive a difficult re-election next year in one of the most critical presidential battlegrounds, she, too, will immediately be nudged to consider a bid.

“She’s been a terrific governor at a very difficult time,” said Senator Debbie Stabenow of Michigan, suggesting Ms. Whitmer could be a strong candidate while also taking care to note that “our vice president is extremely talented.”

Gov. Phil Murphy of New Jersey, having survived a harder-than-expected re-election last month in a dismal political environment, could also run. A onetime Goldman Sachs executive and Democratic donor, he was named ambassador to Germany by former President Barack Obama.

Since his victory, Mr. Murphy has had a series of conversations with prominent Democrats, including a dinner at a well-known New Orleans restaurant with the strategist James Carville that caught the eye of a number of other governors and conference attendees.

There’s also Gov. J.B. Pritzker of Illinois, a billionaire who has worked to stabilize his state’s finances and enact progressive policies, like a $15 minimum wage, since his election in 2018. A longtime financial benefactor of national Democrats, Mr. Pritzker may face a competitive race for re-election in 2022.

While allies say that Mr. Pritzker has expressed no specific intention to run for president in 2024 if Mr. Biden bows out, he has talked privately about his interest in seeking the White House at some point should the opportunity arise.

His advisers tried to tamp down the prospect, at least for now. “Governor Pritzker is focused on addressing the challenges facing the people of Illinois and is not spending any time on D.C.’s favorite parlor game: Who will run for President next,” said Emily Bittner, his spokeswoman. She said the governor “wholeheartedly supports” Mr. Biden and Ms. Harris and expected them to be re-elected.

Still, the talk is abundant — at least in private.

Mr. Trump’s vengeance campaign against Gov. Brian Kemp of Georgia, for example, has Democrats wondering whether Stacey Abrams could take advantage of the Republican disarray to win the state’s governorship and then mount a presidential bid.

Recognizing that such speculation could be used against Ms. Abrams in the governor’s race, her campaign manager insisted last week that if she were elected next year, Ms. Abrams would serve a full term.