Mr. Trump established the commission after his repeated insistence, without credible evidence, that widespread voter fraud explained how Hillary Clinton received about 2.9 million more votes while he won the presidency in the Electoral College.
It is an issue that continues to resonate with his base voters, and Mr. Trump has mentioned it in recent rallies, but there have been few Republicans in Congress who have followed him.
The closing of the commission was a blow for Kris Kobach, the secretary of state of Kansas and the panel’s vice chairman. Mr. Kobach was one of a few state officials to support Mr. Trump’s contention of widespread fraud.
But Mr. Kobach insisted in an interview that the commission’s work would not end but rather would be transferred to the Department of Homeland Security, one of the federal agencies charged with ensuring election integrity and one that he said critics would find more difficult to target.
As a White House commission, the voter-fraud panel was subject to public-disclosure requirements and other restrictions that Mr. Kobach said opponents of the inquiry had seized on in “a determined effort by the left” to hamstring its investigation. At last count, he said, the panel faced at least eight lawsuits accusing it of ignoring various federal requirements, including one from a commission member, Matthew Dunlap, the Maine secretary of state, that claimed he had been illegally excluded from its deliberations.
“It got to the point where the staff of the commission was spending more time responding to litigation than doing an investigation,” Mr. Kobach said. “Think of it as an option play; a decision was made in the middle of the day to pass the ball. The Department of Homeland Security is going to be able to move faster and more efficiently than a presidential advisory commission.”
A spokesman for homeland security, Tyler Q. Houlton, said on Wednesday that “the department continues to focus our efforts on securing elections against those who seek to undermine the election system or its integrity.”
“We will do this in support of state governments who are responsible for administering elections,” he added.
But states may well not cooperate with the department any more than they did with the panel.
As a first step, Mr. Kobach, who said he would remain as an informal adviser to homeland security, said the department would marshal its files on immigrants, legal and otherwise, so that they can be matched with lists of registered voters nationwide to detect foreign citizens who are illegally casting ballots in American elections. Both Mr. Trump and Mr. Kobach have insisted that voting by noncitizens is endemic — Mr. Trump falsely claimed that millions of illegal voters cost him a popular-vote victory in 2016 — but investigations, including ones by Mr. Kobach and the Justice Department under President George W. Bush, turned up scant evidence of fraud.
Many Democratic secretaries of state had said they believed the commission had a goal of laying the groundwork for restrictions that will mostly make it harder for traditional Democratic constituencies — minorities, young people and the poor — to cast ballots, which would benefit Republican candidates.
The commission had been seeking voluminous information on voters, including names, addresses, dates of birth, political affiliations and the last four digits of Social Security numbers, along with voting history. It also had requested records of felony convictions and whether voters are registered in other states.
But many states bar the release of even partial Social Security numbers or other personal information because that data can be used for identity theft.
The commission had faced a deadline days from now about how it would proceed. Vice President Mike Pence, who was tasked with running it, was never particularly excited about the idea, and several members of the commission had objected to working with Mr. Kobach, according to a White House official.
Another official said that the idea, which was supported by the president’s former chief strategist, Stephen K. Bannon, was destined to be shoved off on an agency. And on a day when Mr. Bannon was already under fire for disparaging comments he made in a new book about the presidency, aides put the blame for the existence of the commission on him and insisted he had supported it eagerly. As coverage of the book dominated headlines, the White House pushed out the news of the commission’s closing.
Groups that opposed the commission said its real mission was voter suppression, in ways that would help Republicans, and they were quick to declare victory.
“The commission’s entire purpose was to legitimize voter suppression,” said Vanita Gupta, the president of the Leadership Conference on Civil and Human Rights and former head of the Justice Department’s Civil Rights Division.
“The abrupt abandonment of the commission makes clear that it had become a thoroughly discredited body that could not find evidence of mass voter fraud,” Ms. Gupta said. “The commission itself was unable to justify its existence as a result.”
In a telephone interview late Wednesday, Mr. Dunlap, a Democrat and a member of the panel who has consistently criticized the commission’s operations, said the White House’s decision to move the inquiry to the Department of Homeland Security was “utterly alarming.”
“Homeland security operates very much in the dark,” he said. “Any chance of having this investigation done in a public forum is now lost, and I think people should be, frankly, frightened by that.”
While the conduct of elections now rests with state officials, he said, “Secretary Kobach wants homeland security to make those decisions without public input. That’s the real threat from this decision.”
The Senate Democratic leader, Chuck Schumer of New York, said in a statement that “the commission never had anything to do with election integrity. It was instead a front to suppress the vote, perpetrate dangerous and baseless claims, and was ridiculed from one end of the country to the other.”
Richard L. Hasen, a law professor and election law scholar at the University of California, Irvine, was sharply critical of the commission in a blog post.
“The commission was poorly organized and conceived,” he wrote.
He added, “It made rookie, boneheaded mistakes about handling documents used by the commission, again in violation of federal law. It did not seem to have an endgame.”