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.