Last month, during a virtual debate among the eight candidates running to be Manhattan’s top prosecutor, a final yes-or-no question jolted the group: Would they commit to prosecuting crimes committed by former President Donald J. Trump and his company?

The candidates ducked.

“I actually don’t think any of us should answer that question,” said one contender, Eliza Orlins, as her opponents sounded their agreement.

Despite the candidates’ efforts to avoid it, the question hangs over the hotly contested race to become the next district attorney in Manhattan. The prestigious law enforcement office has been scrutinizing the former president for more than two years and won a hard-fought legal battle this week at the Supreme Court to obtain Mr. Trump’s tax returns.

The current district attorney, Cyrus R. Vance Jr., who has led the office since 2010, is unlikely to seek re-election, according to people with knowledge of his plans, though he has yet to formally announce the decision. He has until next month to decide, but is not actively raising money and has not participated in campaign events.

If Mr. Vance brings criminal charges this year in the Trump investigation, the next district attorney will inherit a complicated case that could take years to resolve. Every major step would need the district attorney’s approval, from plea deals to witnesses to additional charges.

But the most high-profile case in the Manhattan district attorney’s office is also the one that every candidate running to lead the office has been reluctant to discuss.

The eight contenders know that any statements they make could fuel Mr. Trump’s attacks on the investigation as a political “witch hunt,” potentially jeopardizing the case. Many of them have said it is unethical to make promises about Mr. Trump’s fate without first seeing the evidence.

Still, the question comes up repeatedly at debates and forums, a sign of the intense interest surrounding the Trump investigation in Manhattan, where President Biden won 86 percent of the vote in last year’s election.

The candidates are all Democrats, and whoever wins the June 22 primary is almost certain to win the general election in November. At the moment, no Republicans are running. With no public polling available, there is no clear favorite in the race, and in such a crowded field, a candidate may win with a small plurality of the vote. Ranked-choice voting, which will be featured for the first time in the mayoral primary, will not be used in the race.

The candidates have found themselves walking a political tightrope: vowing to hold powerful people like Mr. Trump accountable, without saying too much to prejudge his guilt.

“I’ve been very active and vocal on my feelings on Trump’s abuses of the rule of law, of his terrible policies, of his indecency,” said Dan Quart, a New York State assemblyman who is a candidate in the race. “But that’s different than being a district attorney who has to judge each case on the merits.”

“It’s incumbent upon me not to say things as a candidate for this office that could potentially threaten prosecution in the future,” he added.

The stakes are high. Should Mr. Trump be charged and the case go to trial, a judge could find that the statements made by the new district attorney on the campaign trail tainted the jury pool and could transfer the case out of Manhattan — or even remove the prosecutor from the case, according to legal ethics experts.

Mr. Trump is already laying the groundwork for that argument. In a lengthy statement he released on Monday condemning Mr. Vance’s investigation and the Supreme Court decision, he attacked prosecutorial candidates in “far-left states and jurisdictions pledging to take out a political opponent.”

“That’s fascism, not justice,” the statement said. “And that is exactly what they are trying to do with respect to me.”

Mr. Vance’s investigation has unfolded as a growing number of Democratic leaders have called for Mr. Trump and his family to be held accountable for actions that they believe broke the law.

After the Senate acquitted the former president on a charge of incitement in his second impeachment trial this month, the public interest quickly shifted to the inquiry in Manhattan, one of two known criminal investigations facing Mr. Trump.

Mr. Vance was widely criticized after he declined in 2012 to charge Ivanka Trump and Donald Trump Jr. after a separate fraud investigation and then accepted a donation from their lawyer. The investigation examined whether Trump Organization executives had misled buyers of units at a Trump condo building in Lower Manhattan. (Mr. Vance returned the donation after the public outcry.)

Mr. Vance’s victory over the ex-president at the Supreme Court may temper that criticism. But many of the district attorney candidates have still attacked his decision to close the earlier Trump investigation, campaigning on the belief that his office gave too many free passes to the wealthy and powerful.

In August, Ms. Orlins, a former public defender, suggested on Twitter that, if she were to become district attorney, she would open an investigation into Ivanka Trump.

“You won’t get off so easy when I’m Manhattan D.A.,” she wrote, referring to the fraud investigation that Mr. Vance had shut down. The message drew cheers from her supporters but raised eyebrows among some lawyers.

Erin Murphy, a professor who teaches professional responsibility in criminal practice at New York University School of Law, said the message suggested Ms. Orlins was more focused on a desired outcome than she was on due process.

“It feels like a vindictive thing,” said Ms. Murphy, who supports a rival candidate, Alvin Bragg.

In an interview, Ms. Orlins said that she did not regret the tweet.

“I’m passionate about what I believe,” she said. She maintained that, if elected, she would still evaluate evidence against the Trump family without prejudice.

Some candidates have been more circumspect in addressing the elephant in the room, responding to questions about Mr. Trump by emphasizing their experience investigating powerful people.

Liz Crotty, who worked for Mr. Vance’s predecessor, Robert M. Morgenthau, said in an interview that she would be well-equipped to oversee a complicated case because as a prosecutor she had investigated the finances of Saddam Hussein, the Iraqi dictator.

Diana Florence, a former Manhattan prosecutor, cited her history of taking on real estate and construction fraud to demonstrate that she would not be afraid to pursue the rich and influential.

Mr. Vance’s office began its current investigation into Mr. Trump in 2018, initially focusing on the Trump Organization’s role in hush money payments made during the 2016 presidential campaign to two women who claimed to have had affairs with Mr. Trump.

Since then, prosecutors have suggested in court filings that their investigation has expanded to focus on potential financial crimes, including insurance and bank-related fraud. Mr. Vance has not revealed the scope of his investigation, citing grand jury secrecy.

In August 2019, Mr. Vance’s office sent a subpoena to Mr. Trump’s accounting firm seeking eight years of his tax returns. Mr. Trump repeatedly attempted to block the subpoena. On Monday, the Supreme Court put an end to his efforts, with a short, unsigned order that required Mr. Trump’s accountants to release his records.

Tahanie Aboushi, a civil rights lawyer who is running, said Mr. Vance’s failure to prosecute Mr. Trump earlier reflected a central theme of her campaign. She sees the former president as the beneficiary of a system that allows powerful people to get away with misconduct for which poor people and people of color are harshly punished.

“None of my policies are targeted at Trump or a direct response to Trump,” she said in an interview. “It’s the system as a whole and how it’s historically operated.”

Other candidates have focused on their experience managing complex cases, in tacit acknowledgment of the obstacles ahead in a potential prosecution of a former president. Lucy Lang, a former prosecutor under Mr. Vance running in the race, has touted her familiarity with long-term cases in Manhattan courts, including her leadership of a two-year investigation into a Harlem drug gang.

Daniel R. Alonso, who was Mr. Vance’s top deputy from 2010 to 2014 and is now in private practice, said that any potential case would be an “uphill battle.”

“You can’t have a D.A. who doesn’t have the gravitas and the level of experience to know how to handle the case,” he said.

Several of the contenders already have experience suing the Trump administration and dealing with the scrutiny that comes with it.

Tali Farhadian Weinstein, a former federal prosecutor, has pointed to her role in a lawsuit that successfully stopped federal immigration authorities last year from arresting people at state courthouses. She handled the case as the former general counsel for the Brooklyn district attorney.

Mr. Bragg, who served as a chief deputy at the New York attorney general’s office when it sued Mr. Trump’s charity in 2018, said it was critical in politically charged cases to ignore the public pressure.

“When you do the right thing for the right reason in the right way, justice is its own reward,” he said. “You can’t be motivated by public passions. You have to be rooted in the facts.”