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.