As Gov. Andrew M. Cuomo confronts one of the most seemingly perilous moments in his decade as governor, private conversations are beginning to unfold about what it would take to mount a viable challenge against him next year, and who might be best positioned to take him on.

The New York City public advocate, Jumaane D. Williams, has had conversations with allies in recent weeks about the possibility of seeking higher office. Party insiders hang on every public utterance of the New York attorney general, Letitia James, searching for signs of her future ambitions.

Progressive activists and operatives are trading a flurry of texts, calls and tweets, glued to each fresh controversy unfolding around the governor, and speculating about what the political landscape would look like if he ultimately does not seek a fourth term in 2022. He and his team have said that he intends to run.

Those discussions are in their earliest stages, and in some cases are rooted more in hopes than current realities. But they illustrate a growing sense of uncertainty around Mr. Cuomo, marking a striking turnaround from last year, when some Democrats dreamed of putting him on the presidential ticket.

“Everybody who has ever wanted to be governor has started to go, ‘Oh, what do I need to do if this thing opens up?’” Bill Hyers, a veteran Democratic strategist who managed Mayor Bill de Blasio’s successful 2013 campaign, said this week. “There’s a lined-up coalition who want to defeat him. If he takes two more steps backward, then his challenge will be credible.”

Mr. Cuomo’s administration faces a federal inquiry and legislative backlash in Albany concerning its handling of nursing homes in the pandemic. A number of accusations of bullying behavior have surfaced from lawmakers and former staff members, pushing questions of his temperament into public view.

On Wednesday, a former aide issued a detailed on-the-record accusation of sexual harassment, prompting some officials and New York City mayoral candidates to call for an investigation. His team denies the allegation.

And this week, a Marist poll found that his approval rating had dipped below 50 percent, though other polls have shown him in a much stronger position with Democrats.

The governor’s unsettled future burst into public view at a news conference on Wednesday, where Mr. de Blasio — who has a toxic relationship with Mr. Cuomo and major political challenges of his own — did not rule out a run for governor, in response to a question from The New York Times. A former de Blasio staffer recalled that several years ago, the mayor would mention the idea of challenging Mr. Cuomo as one eventual possibility (though the mayor has middling approval numbers in his own city and little demonstrated support outside it).

On the Republican side, Representative Tom Reed, a co-chairman of the bipartisan Problem Solvers Caucus, has said that he is “seriously considering” a run, and party activists and officials have mentioned other possible contenders, including Representatives Elise Stefanik from the North Country and Lee Zeldin from Long Island. Their records of strong support for former President Donald J. Trump would be a major liability in a statewide race.

National Republicans are aware of the challenges of running in heavily Democratic New York, barring significant weakening of Mr. Cuomo, but they are watching possibilities for that race closely, they say.

Any serious threat to Mr. Cuomo would be more likely to emerge in a Democratic primary.

“Any elected official that does not respond to the mandate of the people deserves a primary, myself included,” State Senator Alessandra Biaggi, a sharp critic of Mr. Cuomo’s, said in an interview late last week. She said she did not believe Mr. Cuomo had responded to that mandate.

“Right now, his leadership is not hitting the mark,” she said. “And I think that New York deserves the best leadership, and we don’t have any more time. We are out of time, with below-average leaders who refuse to have integrity.”

Asked if she would consider running for governor herself, she replied, “No, not today.” As for next year, she insisted that “that is not even what I’m concerned with right now,” as she navigates the needs of her Bronx and Westchester district, which has been hit hard by the pandemic.

Certainly, even Mr. Cuomo’s biggest detractors are cleareyed about just how difficult it would be to challenge the governor.

He defeated his last two primary opponents by around 30 percentage points each. He is a ruthless campaigner with a huge war chest and a lengthy record of achievements, and he has significant strength in communities of color. Many New Yorkers harbor good will toward him for his efforts to reassure the state in the early months of the pandemic, and it is unclear how much the turbulence of recent weeks resonates with voters now, much less how it will play out next year.

“In the past there have been many challenges that started with perceived fanfare, that ended in a fizzle,” said Jefrey Pollock, Mr. Cuomo’s pollster. “The governor’s record as one of the most progressive governors in the country is the thing that’s going to carry him to re-election, to first a primary victory and then victory in the general.”

Democratic strategists eager to challenge Mr. Cuomo have a particular focus on who could connect with Black voters, a constituency that has been vital to Mr. Cuomo’s success in the past.

Ms. James, who released a major report about how the Cuomo administration undercounted nursing home resident deaths tied to Covid-19, is the first Black woman to hold statewide office. She has led a number of progressive charges in office, and she has generated significant discussion among liberal leaders and strategists.

People who have known her over the years see her as politically risk-averse and are skeptical, at this point, that she would challenge Mr. Cuomo, who has been a key ally.

But a number of strategists note that the position of attorney general has often been a launching point for governor — as it was for Mr. Cuomo — and believe she would be formidable if he was not running, or if there are drastic changes to his political fortunes. A spokeswoman for Ms. James declined to comment.

“I do know that there are others who say that attorney general stands for ‘aspiring governor,’’’ Ms. James said in an interview with The New York Times DealBook/DC Policy Project this week, saying she did not view her role as a “political job.”

“At this point in time, my focus, again, is representing the interests of the citizens of the great State of New York,” she said.

Then there are a number of prominent progressive state legislators who have clashed with Mr. Cuomo for years, like Ms. Biaggi, who has been an especially visible critic of his handling of the nursing homes controversy.

State Senator Jessica Ramos, another Cuomo critic, is also mentioned in some circles as a potential contender. But in a phone call on Thursday, she alluded to the significant financial hurdles any challenger would face.

“We definitely need a true progressive governor, and I would love to see working people in New York coalesce around one candidate,” she said. “The part I think is a very serious challenge is when it comes to fund-raising, when we’re trying to represent those who have nothing.”

Some progressives are also discussing the future of Mr. Williams, the public advocate, who is running for re-election this year but has spoken with allies about the possibility of running for governor or lieutenant governor. Mr. Williams, who lost a 2018 bid for lieutenant governor by around 6.6 percentage points, has been thought to be more likely to pursue that post again if he runs for another office. The Rev. Al Sharpton, the civil rights leader, said that he cornered Mr. Williams on the subject ahead of an event honoring the Rev. Martin Luther King Jr.

“In my office, getting ready to go out, I said to Jumaane, ‘You thinking of running for lieutenant governor again?’” Mr. Sharpton said. “He just smiled, didn’t deny, didn’t agree.”

But allies have reached out to Mr. Williams in recent weeks about running for governor, too, according to a person familiar with the conversations.

“The activists and folks like us would be very excited to see someone like Jumaane Williams run for governor,” said Jonathan Westin, who leads the progressive group New York Communities for Change. “If he ran, he could really give him a run for his money in a lot of Black and brown neighborhoods across New York.”

Many strategists, officials and others looking at the race don’t expect the potential primary field to take shape for some months, and Mr. Cuomo’s many defenders across the party remain bullish on his chances.

“Anybody is vulnerable, but anybody who primaries him does so at their peril,” said Keith L.T. Wright, the leader of the New York County Democrats.

But as scrutiny over Mr. Cuomo’s behavior mounts, there remains the broader question of whether others in Albany may begin to turn on him — as they turned on ex-Gov. Eliot Spitzer when he was in the midst of a scandal — or if ultimately many Democrats will close ranks.

“In order for him to be vulnerable, you’ve got to come with the candidate,” Mr. Sharpton said. “There is a lot of bad press, but I don’t see the candidate.”

Dana Rubinstein contributed reporting.