Nearly eight months ago, Scott M. Stringer stood in Brooklyn before an angry, unsettled gathering to memorialize the death of George Floyd. The best way to honor him, Mr. Stringer said, was to send a clear message to City Hall: “It’s time to defund the N.Y.P.D. now.”

But with the New York City mayoral primary looming in June, Mr. Stringer has distanced himself from the defund movement.

At a recent mayoral forum, Mr. Stringer was asked if he supported defunding the police, and whether he would commit to slashing the Police Department’s $6 billion budget in half. He responded with a less drastic proposal to cut $1 billion, spread over four years, and said he did not want the city to return to a period of high crime like in the 1970s.

“I do remember when the A train was a rolling crime scene, and I don’t want my children or any child to go back to that time and place,” he said. “But I also know that overpolicing in communities of color has got to stop.”

His remarks immediately drew fire.

“This is not what a progressive would say,” said Lauren Ashcraft, a Democratic Socialist and former congressional candidate in Queens. Other progressives questioned whether several leading Democratic candidates had the courage and commitment to win their support.

The escalating tensions over the issue highlight the challenges that Democratic candidates face as they try to cultivate the city’s growing progressive flank without embracing stances that may scare off moderate New Yorkers — especially at a time when shootings and murders have sharply risen.

The issue cuts across racial and class lines: Two Black moderate Democratic candidates, Eric Adams and Raymond J. McGuire, have voiced concerns — echoed among other Black lawmakers in the city — that defunding the police would worsen crime in neighborhoods that suffer the most from violence.

Maya Wiley, a former top counsel for Mr. de Blasio who gained a national following as an analyst for MSNBC, was often critical of the mayor’s handling of policing. Now she appears to be recalibrating her message to avoid using the defund slogan.

“The word means different things to different people,” Ms. Wiley said. “We should focus on the clarity of the demands.”

Others in the wide-open Democratic field for mayor have sought to distance themselves from the defund movement and instead speak more of the need to bring meaningful change to the Police Department.

The debate over the defund movement has roiled the Democratic Party over concerns that the slogan scared away moderate voters during the election in November. Some Democratic leaders blamed candidates’ embrace of the movement for the party’s losses in the House.

President Barack Obama discouraged candidates from using the slogan — arguing you have “lost a big audience the minute you say it” — while leaders on the progressive left like Representative Alexandria Ocasio-Cortez defended it and blamed moderates for running weak campaigns.

Even in Minneapolis, where Mr. Floyd was killed by the police, efforts to essentially dismantle the city’s police department collapsed. A far less ambitious move — cutting the police budget by 4.5 percent — was approved in December, disappointing defund supporters.

In a survey of the nine leading Democratic mayoral hopefuls, only two said they supported the defund movement: Dianne Morales, a former nonprofit executive, and Carlos Menchaca, a councilman from Brooklyn.

The race for mayor this year may be the city’s most consequential in a generation, with New York facing a resurgence of the coronavirus that may prolong and worsen the city’s economic crisis. The pandemic’s effect on the city has overshadowed other issues on the campaign trail, including defunding the police.

Many of the campaigns have commissioned polls to measure which issues voters want the next mayor to prioritize, and they have consulted with policing experts about how to tackle reforms. Keeping New Yorkers safe from the pandemic was the top concern in one poll; defunding the police was not among the Top 10 issues, with voters caring more about keeping crime down.

“A lot of people, including in communities impacted by policing, bristle at the term,” Barry Friedman, a professor and director of the Policing Project at New York University School of Law, said of the defund slogan. “There are people who are frustrated at how police respond to situations, but don’t think they’re going to be safer without the police.”

Still, many left-leaning leaders in New York are committed to trying to keep defund efforts alive. Two advocacy groups and one union — Make the Road Action, Community Voices Heard Power and 1199 Service Employees International Union — plan to unveil an independent expenditure committee to make the defund movement one of the top issues in the mayor’s race.

The New York City Democratic Socialists of America, which helped Ms. Ocasio-Cortez win her primary in 2018, intends to make a concerted push to make defunding the police a key issue, according to the group’s co-chairwoman, Sumathy Kumar.

Tiffany Cabán, a progressive-backed City Council candidate who nearly scored an upset win in the 2019 Democratic primary for Queens district attorney, wrote a 40-page public safety platform that is expected to be the philosophical basis that defund supporters running for mayor or City Council organize around.

Ms. Cabán has spoken with Ms. Morales, Ms. Wiley and Mr. Stringer about what proponents of defunding the police want. She said she often had to tell candidates that their positions did not go far enough and believes that will change.

“There will be a domino effect,” she said. “One of the big, more forward-facing mayoral candidates is going to release their comprehensive police plan, and everyone else is going to have an answer to it.”

Mr. Stringer, who has won endorsement from several progressive leaders who support the defund movement, said that he was “the first elected official to put forth a detailed proposal to reduce the N.Y.P.D.’s budget by $1 billion,” and that his “position on these issues has not changed.”

But when asked directly whether he supports defunding the police, Mr. Stringer gave a more indirect answer, saying that he wanted to “make concrete change when it comes to systemic racism and our criminal justice system.”

Mr. Stringer will soon release a report that explains his policy ideas to transform policing that he says is more ambitious than his proposal in June. The report, a copy of which was reviewed by The New York Times, does not mention the word “defund” or cite a dollar figure for budget cuts, but it outlines how he would move certain responsibilities away from the Police Department and identifies specific areas for reinvestment in communities.

Most of the other major candidates seemed even less likely to make major cuts to the police budget.

Mr. McGuire, who served on the New York City Police Foundation, has called for “better policing with greater accountability, not fewer police officers,” even as he has said that as a Black man he could “easily be the next George Floyd” — a contrast that he does not view as a contradiction.

Mr. Adams, a former police officer, said that he does not “support taking resources away from crime fighting — especially in communities of color where shootings and other predatory crimes are on the rise.”

Ms. Wiley said at a recent mayoral forum that the Police Department budget was “bloated,” but declined to say how much she would seek to cut police spending.

“I don’t have a number for you, but that’s because it has been such a black box,” Ms. Wiley said. “There really is so little transparency about what and how the budget is spent.”

Kathryn Garcia, the city’s former sanitation commissioner, has called for “police reform through strict accountability and structural change,” including raising the minimum age of recruits to 25 and holding “police officers accountable for depraved acts with a zero-tolerance policy.”

Shaun Donovan, the former federal housing secretary and budget director, called for a reduction in “overpolicing” and reinvestment in “wraparound social services.”

Andrew Yang, a former presidential candidate, said the Police Department needs resources to address rising crime, but he supported shifting some of its funding to other city agencies that could better handle certain issues.

“Not every problem requires an armed police officer,” Mr. Yang said.

Last year was New York City’s bloodiest in nearly a decade with more than 460 homicides; the number of shooting victims doubled to more than 1,500. Mr. de Blasio and police leaders have blamed the economic losses and upheaval of the pandemic.

Murders surged in Black and Latino neighborhoods including East Harlem, East New York in Brooklyn, Rockaway Peninsula in Queens, and areas near Yankee Stadium in the Bronx. In the 73rd Precinct in Brooklyn, which includes Brownsville and where about 70 percent of residents are Black, there were 25 murders last year, compared with 11 in 2019.

The June 22 Democratic primary will take place just days before the City Council’s deadline to approve the budget for the next fiscal year, all but assuring that the push to defund will be in the public and political discourse.

Last June, the defund effort led the city to pass a budget that called for the Police Department to suffer “$1 billion in cuts and cost shifts,” according to the mayor. But an analysis by the Independent Budget Office concluded that a smaller portion of the police budget was actually cut, and some of the losses were spread out over a number of years.

Mr. Menchaca voted against that budget because it failed to cut $1 billion from the Police Department; he said he plans to raise the issue during upcoming Council budget negotiations.

The defund movement, he said, “is going to be on the ballot.”

Ms. Morales said she reached out to the Brooklyn Movement Center last summer to get a sense of what defund advocates expected from the next mayor. She now wants to cut $3 billion from the police budget — a position that has won her support among defund advocates.

“We need to take that money and invest it in meeting people’s needs,” she said at a recent mayoral forum.

Anthonine Pierre, deputy director of the Brooklyn Movement Center and a member of Communities United for Police Reform, one of the architects of the city’s defund movement, said that she was not surprised that more mainstream candidates have not reached out to her group because they think of “defund the police” as a communications strategy.

“Scott Stringer has had over a two-decade career in New York City politics, and never has police accountability been a banner issue for him,” said Ms. Pierre, who worked for Mr. Stringer in 2008 when he was the Manhattan borough president.

“There is really a lack of courage from these candidates,” she said, adding that she would welcome discussions with the mayoral field — or a direct message on Twitter.

“My DMs are open,” she said.