As they investigate former President Donald J. Trump’s efforts to overturn the 2020 election, federal prosecutors have also been drilling down on whether Mr. Trump and a range of political aides knew that he had lost the race but still raised money off claims that they were fighting widespread fraud in the vote results, according to three people familiar with the matter.

Led by the special counsel Jack Smith, prosecutors are trying to determine whether Mr. Trump and his aides violated federal wire fraud statutes as they raised as much as $250 million through a political action committee by saying they needed the money to fight to reverse election fraud even though they had been told repeatedly that there was no evidence to back up those fraud claims.

The prosecutors are looking at the inner workings of the committee, Save America PAC, and at the Trump campaign’s efforts to prove its baseless case that Mr. Trump had been cheated out of victory.

In the past several months, prosecutors have issued multiple batches of subpoenas in a wide-ranging effort to understand Save America, which was set up shortly after the election as Mr. Trump’s main fund-raising entity. An initial round of subpoenas, which started going out before Mr. Trump declared his candidacy in the 2024 race and Mr. Smith was appointed by Attorney General Merrick B. Garland in November, focused on various Republican officials and vendors that had received payments from Save America.

But more recently, investigators have homed in on the activities of a joint fund-raising committee made up of staff members from the 2020 Trump campaign and the Republican National Committee, among others. Some of the subpoenas have sought documents from around Election Day 2020 up the present.

Prosecutors have been heavily focused on details of the campaign’s finances, spending and fund-raising, such as who was approving email solicitations that were blasted out to lists of possible small donors and what they knew about the truth of the fraud claims, according to the people familiar with their work. All three areas overlap, and could inform prosecutors’ thinking about whether to proceed with charges in an investigation in which witnesses are still being interviewed.

The possibility that the fund-raising efforts might have been criminally fraudulent was first raised last year by the House select committee investigating Mr. Trump’s efforts to retain power.

But the Justice Department, with its ability to bring criminal charges, has been able to prompt more extensive cooperation from a number of witnesses. And prosecutors have developed more information than the House committee did, having targeted communications between Trump campaign aides and other Republican officials to determine if a barrage of fund-raising solicitations sent out after the election were knowingly misleading, according to the three people familiar with the matter.

The fund-raising efforts are just one focus of Mr. Smith’s investigation into Mr. Trump’s attempts to reverse his loss at the polls.

Prosecutors have also been examining the plan to assemble alternate slates of pro-Trump electors from swing states won by Joseph R. Biden Jr., and the broader push by Mr. Trump to block or delay congressional certification of Mr. Biden’s Electoral College victory on Jan. 6, 2021, leading to the storming of the Capitol by Trump supporters.

On Thursday, former Vice President Mike Pence, a key witness to Mr. Trump’s efforts, testified for hours to the grand jury gathering evidence in the investigation.

Prosecutors have been looking at the nexus between research the Trump campaign commissioned almost immediately after the election to try to prove widespread fraud, public statements that he and his allies made at the time, the fund-raising efforts and the establishment of Save America.

The Washington Post reported earlier on the efforts by the campaign to fund research into claims of fraud and the new round of subpoenas.

Mr. Trump’s team may argue that the fund-raising represented political speech with solicitations that were generally vague, and that subjecting it to a criminal process could raise First Amendment issues and create a slippery slope for future candidates. Political fund-raising materials often engage in bombast or exaggeration.

Republicans may also argue that Democrats have been loose in claims they have used in fund-raising solicitations. And the Trump campaign may argue that it did in fact use the funds to try to investigate fraud.

A Trump campaign adviser said the “deep state” was ramping up its attacks on the former president as his poll numbers rose. “The ‘political police’ have been pushing their witch hunt since President Trump came down the escalator, and they’ve been proven wrong every single time,” the adviser added.

Officials with the Republican National Committee declined to comment.

Immediately after the election, an adviser to the Trump campaign reached out to Ken Block, the owner of a Rhode Island-based firm, Simpatico Software Systems, to have him evaluate specific allegations of fraud.

Mr. Block ended up researching multiple claims of possible fraud that Mr. Trump’s aides brought to him. He never produced a final report. But each time he investigated a claim, he said in an interview, he found there was nothing to it.

Mr. Block said he had disproved “everything that came in and found no substantive fraud sufficient to overturn an election result.” He said he was isolated from what was taking place within the campaign, as Mr. Trump railed at aides about staying in office and continued to insist he had won an election that he was repeatedly told he had lost.

“I was kept very walled off from all of the insanity,” said Mr. Block, whose firm was paid $735,000, records show. He received a subpoena for documents, but declined in the interview to discuss anything related to the grand jury.

Days after starting to work with Mr. Block and Simpatico, the Trump campaign hired a second firm, the Berkeley Research Group. The federal grand jury has received evidence that Berkeley was hired at the suggestion of Jared Kushner, Mr. Trump’s son-in-law, who was overseeing the political operation.

The grand jury has been asking questions related to whether Mr. Trump was briefed on findings by Berkeley suggesting there had been no widespread fraud.

The company ultimately submitted a report indicating there had been no fraud that would have changed the outcome of the election, and was paid roughly $600,000 for its work. The company was hired through a law firm that has long represented Mr. Trump in his personal capacity, Kasowitz Benson Torres, although lawyers there were not involved in pursuing Mr. Trump’s election fraud claims, according to a person briefed on the matter.

A deputy counsel for Berkeley Research Group said the company has a “no comment” policy and declined to discuss the matter further.

During the House Jan. 6 committee’s proceedings last year, several people close to Mr. Trump testified that they had informed him that there had been no fraud sufficient to change the outcome of the voting.

Within two weeks of the election, the Trump campaign’s own communications staff drafted an internal report debunking many aspects of a conspiracy theory that voting machines made by Dominion Voting Systems had been hacked and used to flip votes away from Mr. Trump. That report was written before pro-Trump lawyers like Sidney Powell and Rudolph W. Giuliani promoted the false Dominion story at news conferences and on television.

As part of its investigation into the Trump campaign’s postelection fund-raising, the Jan. 6 panel subpoenaed records from Salesforce.com, a vendor that helped the campaign and the Republican National Committee send emails to potential donors. The R.N.C. fought back, filing a lawsuit to quash the subpoena, and the House committee ultimately withdrew it.

In the latest round of subpoenas, federal prosecutors have sought documents related to Salesforce in addition to other vendors, according to a person briefed on the matter.