A White House commission’s sweeping request for the personal and public data of the nation’s 200 million voters set off an avalanche of opposition by state leaders in both parties on Friday, as officials from California to Mississippi called the move an overreach and more than 20 states declared they would not comply.

It was an inauspicious start for the panel, which was created after President Trump claimed last winter that millions of illegal votes had robbed him of a popular-vote victory over Hillary Clinton.

The vice chairman and day-to-day leader of the Presidential Advisory Commission on Voter Integrity, Kris Kobach, had asked election officials in a letter to turn over the data “if publicly available,” apparently to aid a nationwide search for evidence of election irregularities. Besides election information like voters’ names and party affiliations, the commission sought personal information including birth dates, felony conviction records, voting histories for the past decade and the last four digits of all voters’ Social Security numbers.

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Mr. Kobach, the secretary of state in Kansas, has said he wants to match voter information with other data, like federal records of foreign residents and undocumented immigrants, to spotlight people who cast illegal ballots. He asserts that such fraud is widespread in Kansas and elsewhere, although he has found scant evidence of it so far.

But a growing number of state election officials have indicated — sometimes politely, sometimes brusquely — that they will not or cannot comply. Among them, ironically, were Mr. Kobach himself and a second member of the commission, Secretary of State Connie Lawson of Indiana, both of whom disclosed on Friday that privacy laws prevented them from furnishing some personal voter data.

By Friday, an informal tally by voting-rights advocates indicated that election officials in at least 22 states had partly or completely rejected the commission’s request.

California, Massachusetts, Virginia, New York and Kentucky all quickly rejected the request. Other states, like Connecticut and Tennessee, said state law barred them from turning over some data. Wisconsin pledged to provide what it legally could, if the commission paid the $12,500 fee charged to anyone who copies the voter rolls.

Kentucky’s secretary of state, Alison Lundergan Grimes, said that Mr. Trump’s premise for creating the commission in the first place — that voter fraud was pervasive and needed to be reined in — was itself a fraud.

“Kentucky will not aid a commission that is at best a waste of taxpayer money and at worst an effort to legitimize voter suppression efforts across the country,” Ms. Grimes, a Democrat, wrote in response to Mr. Kobach’s request.

The pushback was bipartisan: The Mississippi secretary of state, Delbert Hosemann, a Republican, said Friday that he had not received a request from the commission, but colorfully suggested he would not honor one if it came.

“My reply would be: They can go jump in the Gulf of Mexico, and Mississippi is a great state to launch from,” Mr. Hosemann said in a statement. “Mississippi residents should celebrate Independence Day and our state’s right to protect the privacy of our citizens by conducting our own electoral processes.”

Mr. Kobach, whose spokeswoman did not respond to phone and email messages, told The Kansas City Star on Friday that he was not concerned by other states’ refusals to disclose voters’ personal data. “That’s perfectly fine,” the newspaper quoted him as saying. “We understand that. And that is entirely up to each state.”

In an interview last week with The Washington Times, Mr. Kobach said the accusations from voting-rights advocates and Democrats that the commission is a pretense for a voter-suppression enterprise designed to benefit Republicans were “complete and utter nonsense.” Mr. Kobach told the newspaper that the act of collecting data posed no threat to voters, saying that the commission intended to match voter rolls with the federal government’s database of noncitizens — including permanent residents, undocumented immigrants who had been apprehended and others — in a search for fraudulent ballots.

Much of the voter data sought by the commission — which is formally led by Vice President Mike Pence, as its chairman — is either public information or is routinely provided to political parties, researchers and others. But at least in California, some of it is protected by law from disclosure, said Alex Padilla, the secretary of state. And the personal data sought by the commission has never been aggregated on a national level with voting information, and should not be, he said.

Beyond concerns about privacy and how the data would be used, said Mr. Padilla, a Democrat, “I don’t want Kris Kobach to do to California what he’s done to Kansas.”

Mr. Kobach, a Republican, has claimed voter fraud is rampant in Kansas, particularly by unauthorized immigrants. He has pushed for an array of restrictions on voting and registration — some of them overturned after legal battles — that a federal judge said had kept thousands of Kansans off the rolls.

Academic research and reports by the states themselves have repeatedly concluded that voter fraud is exceedingly rare and limited largely to absentee ballots and vote-rigging by election officials. But some scholars and advocates suspect that the commission’s true goal is to paint a portrait of a voting system vulnerable to manipulation, regardless of the actual prevalence of fraud.

“I think the endgame here is to provide the pretext for federal legislation to make it harder for people to register and vote,” Richard L. Hasen, a professor and elections expert at the University of California, Irvine School of Law, said in an interview.

Mr. Hasen and some voting-rights advocates say they anticipate that the commission’s findings will be followed by Republican legislation to amend the 1993 National Voter Registration Act — the so-called motor voter law, a longtime target of Republican complaints.

New legislation could give states more leeway to impose voting preconditions like proof of citizenship and to make it easier for officials to purge inactive voters from the rolls, Mr. Hasen said.

The commission is officially bipartisan, and Mr. Kobach has said he will bring no biases to the job. That said, the Republicans named to the panel so far include some of the most ardent advocates of voting restrictions, led by Mr. Kobach and J. Kenneth Blackwell, who pushed for more stringent voting rules in the mid-2000s as Ohio secretary of state.

Another Republican appointee announced on Thursday, Hans A. von Spakovsky, is a former Justice Department official in the George W. Bush administration who advocates stricter laws on voting and registering. Mr. von Spakovsky, a Heritage Foundation scholar who maintains an online repository of voting-fraud convictions, has crusaded against what he calls a liberal bias in federal enforcement of election laws. “When it comes to American elections, the Obama Justice Department has taken the side of criminals over law-abiding citizens,” he wrote last year in a critique of the department’s lawyers. “It has used selective enforcement of the law to help Democrats retain power.”

Voting-rights advocates have urged Democrats to shun membership on the commission, but Maine’s secretary of state, Matthew Dunlap, agreed to join it this month — in part, he said in an interview, to serve as a watchdog.

“If it takes a dishonest turn, say, and instead of having our thesis informed by facts we have our facts informed by the thesis, I’m in an incredible position to talk about that publicly,” Mr. Dunlap said.