Prosecutors in the office of the special counsel Jack Smith are pressing forward with their investigation into former President Donald J. Trump’s attempts to overturn the 2020 election, asking at least one witness this week about fund-raising efforts by Mr. Trump’s political action committee.

The witness, Bernard B. Kerik, sat for a voluntary interview on Monday with prosecutors in what is known as a proffer session. A former New York City police commissioner, Mr. Kerik was instrumental in helping Rudolph W. Giuliani, a former mayor of New York and one of Mr. Trump’s chief lawyers after the election, investigate claims of fraud in the vote results.

Mr. Kerik’s interview was first reported by Politico.

Among the questions prosecutors asked Mr. Kerik were several related to Mr. Trump’s main postelection fund-raising entity, Save America PAC. The special counsel’s office has been drilling down for months into whether the political action committee raised millions of dollars on claims that there was widespread fraud in the election, but ultimately earmarked the money for things other than investigating those claims.

Mr. Kerik told prosecutors that the team Mr. Giuliani had assembled to look into the allegations of fraud received no money from Save America PAC, even though it was one of the chief groups assigned the task of hunting down evidence that the election had been marred by cheating, Mr. Kerik’s lawyer, Timothy Parlatore, said on Tuesday.

Mr. Kerik also told prosecutors that if Save America had provided money to Mr. Giuliani’s team, it might have more accurately vetted the claims of fraud, Mr. Parlatore said.

Proffer sessions are based on an agreement between prosecutors and people who are subjects of criminal investigations. The subjects agree to provide useful information, sometimes to tell their side of events, to stave off potential charges or to avoid testifying under subpoena before a grand jury. In exchange, prosecutors agree not to use those statements against them in future criminal proceedings unless it is determined they were lying.

Mr. Giuliani sat for his own proffer session over two days in June and was asked several questions about a key aspect of Mr. Smith’s investigation: a plan by Mr. Trump and his allies to create fake slates of pro-Trump electors in key swing states that were actually won by President Biden. Prosecutors were said have to focused specifically on the role played in that effort by John Eastman, another lawyer who advised Mr. Trump about ways to stay in office after his defeat.

Both Mr. Giuliani and Mr. Eastman appeared as unnamed co-conspirators in the indictment that the special counsel filed against Mr. Trump in Washington this month. The indictment accused the former president of three overlapping conspiracies to defraud the United States, to disrupt the certification of the election at a joint session of Congress on Jan. 6, 2021, and to deprive people of the right to have their votes counted.

Judge Tanya S. Chutkan, who is presiding in the case, has set her first hearing in the matter for Friday morning to discuss imposing a protective order regulating how Mr. Trump and his lawyers should handle the voluminous discovery evidence they are about to receive from the government.

Prosecutors have asked Judge Chutkan to bar Mr. Trump from publicly releasing any of the evidence out of concern that he will seek to try the case in the media before it reaches the courtroom.

Mr. Trump’s lawyers have asked for a narrower order, arguing that the restrictions sought by the government infringe on Mr. Trump’s First Amendment rights to discuss the case as he runs for public office.

On Tuesday evening, Mr. Trump’s lawyers filed court papers that also asked Judge Chutkan to exclude the 25 days between Mr. Trump’s arraignment on Aug. 3 and the first status conference in the case on Aug. 28 from what is known as the speedy trial clock.

By law, defendants are supposed to go to trial within 70 days of being charged — unless the judge in the case subtracts time from that period. Mr. Trump’s lawyers urged Judge Chutkan to do so, arguing that the complexity of the proceeding demanded it.

“For a case of this magnitude, it would be impossible for the defense to evaluate the government’s evidence, prepare its own defense and participate in pretrial proceedings, all within the time constraints of the Speedy Trial Act,” the lawyers wrote.

One of Mr. Trump’s chief legal strategies in both the election interference case and the case related to his handling of classified documents has been to delay going to trial for as long as possible. If he is able to push the trials until after the 2024 election and prevails in the race, he could either seek to pardon himself or have his attorney general simply dismiss the charges.

During Mr. Kerik’s interview on Monday, prosecutors asked about another person indicated as a co-conspirator in Mr. Trump’s indictment: Sidney Powell, a Texas-based lawyer who filed a series of conspiracy-laden lawsuits after the election claiming that voting machines manufactured by Dominion Voting Systems had been used in plot to flip votes away from Mr. Trump to Mr. Biden.

Prosecutors asked Mr. Kerik on what factual basis he believed Ms. Powell had filed her suits and he responded that he was unaware of one.

Prosecutors also asked Mr. Kerik about Phil Waldron, a former Army colonel from Texas who served as a kind of liaison between Ms. Powell and members of Mr. Giuliani’s team. Mr. Smith’s investigators wanted to know how seriously Mr. Kerik and others on the team had vetted Mr. Waldron’s claims that there were mathematical irregularities in the vote results in some key swing states that indicated fraud, Mr. Parlatore said.