The Presidential Advisory Commission on Election Integrity, formed last month via an executive order by President Trump, asked the 50 secretaries of state this week to provide the voter data, The Hill and The Washington Post report.
Among the states objecting so far are California, Connecticut, Indiana, Iowa, Kentucky, Massachusetts, Maine, Minnesota, Mississippi, New Mexico, Nevada, New York, North Dakota, Ohio, Oklahoma, Pennsylvania, Rhode Island, Tennessee, Utah, Vermont, Virginia, Washington and Wisconsin.
Kentucky Secretary of State Alison Lundergan Grimes said she would not provide the data.
"The president created his election commission based on the false notion that 'voter fraud' is a widespread issue — it is not," Lundergan Grimes, a Democrat, told the Hill. "I do not intend to release Kentuckians' sensitive personal data to the federal government."
Jon Husted, Ohio's Republican secretary of state, said that in the Buckeye State, "we pride ourselves on being a state where it is easy to vote and hard to cheat.
"Voter fraud happens. It's rare and when it happens we hold people accountable.
Alex Padilla, the Democratic secretary of state in California, told the Hill in a statement: "California's participation would only serve to legitimize the false and already debunked claims of massive voter fraud."
President Trump has alleged that millions of illegal votes cost him the popular vote, though he has offered no evidence to support his claim.
Numerous state and academic studies of voter files show that only a handful of improper votes were cast in recent elections, the Hill reports.