A bipartisan group of 35 former federal judges on Wednesday asked the judge who oversaw President Trump’s remarkable lawsuit against the Internal Revenue Service to reopen the case and conduct an inquiry into whether the hasty deal to resolve it could be challenged as an act of fraud.

The move by the former judges was one of an increasing number of legal efforts to attack the validity of the two extraordinary benefits that emerged from the agreement last week: a $1.8 billion fund that could compensate allies of Mr. Trump who claim they suffered “weaponization” at the hands of the federal government and the conferral of lucrative tax benefits on the president, his family and his businesses.

The motion by the former judges, filed in Federal District Court in Miami, was a direct appeal to Judge Kathleen M. Williams, who closed the I.R.S. case last week after Mr. Trump voluntarily dismissed his suit. It asked her to bring the matter back to life under a rule that permits her to set aside a judgment she had made and examine the terms of the deal that appeared to have been reached in a plan to avoid that sort of scrutiny.

“The purported ‘settlement’ that was publicly disclosed after this court dismissed this matter raises profound questions about the parties’ candor toward the court and manipulation of the judicial system, which threatens to undermine confidence in the administration of justice,” lawyers for the former judges wrote.

The motion came as the fund faced stiff headwinds on Capitol Hill, especially among Senate Republicans who expressed deep skepticism about a pot of taxpayer money that could be used to issue payouts to Trump supporters, including the rioters prosecuted for attacking the Capitol on Jan. 6, 2021.

At the heart of the former judges’ argument was an assertion that Mr. Trump improperly used his lawsuit against the I.R.S. as a way to obtain “unlawful private benefits” for himself and his family and to create a fund that would dole out taxpayer money “without constitutional or congressional authority.” Moreover, the former judges claimed that the president tried to shield the deal from “judicial scrutiny” by “short-circuiting” Judge Williams’ ability to examine its terms.

Mr. Trump, along with two of his sons and the Trump family business, first sued the I.R.S. in January, claiming they were owed at least $10 billion because a former contractor at the agency had leaked their tax returns (and hundreds of others’) during the president’s first term in the White House. The Trumps claimed that the I.R.S. should have done more to prevent the contractor, Charles Littlejohn, from disclosing tax information to The New York Times and ProPublica.

But as Judge Williams dug into the suit, she began to have qualms about whether it was valid, given that Mr. Trump — by his own admission — was engaged in a form of legal self-dealing. The judge wondered whether there was an actual conflict at stake that she could adjudicate if the president was essentially on both sides of the suit, seeking damages from a federal agency that he directly controlled.

To get to the bottom of that question, Judge Williams ordered the parties in the suit to explain their position on the case. But just two days before those briefs were due, Mr. Trump abruptly withdrew his suit, telling the judge that she had no authority to question the dismissal because the I.R.S. had never formally responded to the claims.

Acknowledging that her hands were tied, Judge Williams quickly closed the case, but noted in her order that there had never been a “settlement of record.” Within hours, however, the terms of a deal surfaced in public in an agreement that was signed by a senior Justice Department official and detailed how the fund would work. The next day, the department released an addendum to the agreement giving the Trump family its own extraordinary boon: immunity from all past I.R.S. investigations.

In their court papers, the former judges laid out that sequence of events, telling Judge Williams that it showed the fraudulent nature of the agreement. The judges said she did not have to rule immediately that settlement was invalid, suggesting that she could first “commence an inquiry into the whether the court was deceived.”

The judges reminded Judge Williams that the Justice Department had fought on behalf of the I.R.S. against nearly identical claims that were brought by the other people whose tax returns had been leaked by Mr. Littlejohn. The I.R.S. had even written a 25-page memo, they said, laying out an unused plan to defend against Mr. Trump’s claims.

In recent days, two other lawsuits have been filed seeking to block the compensation fund from making any payouts.

One of the suits was filed by two Capitol Police officers who were injured while defending lawmakers on Jan. 6 and claimed that the Trump administration had exceeded its statutory authority by setting up the fund without congressional authorization.

The other was filed by a former federal prosecutor who was fired after working on Jan. 6-related cases, a California college professor who was arrested while protesting an immigration raid, and the city of New Haven, which was targeted by the Trump administration for being a “sanctuary” city for immigrants. That suit claimed the fund violated the First Amendment and the Constitution’s equal protection clause because of the “blatant partiality” it showed in making money available to people who claimed to have been targeted by Democratic administrations.

Still, the plaintiffs in those suits could run into trouble showing they have legal standing to file their claims.