Mr. Trump’s words, though, appear to be having an effect on his supporters, and are setting off deep concern among civil rights groups. According to an Associated Press poll last month, only one-third of Republicans said they had a great deal of confidence their votes would be counted fairly. And election officials are worried that Mr. Trump’s continued pressing of the issue could dampen turnout or cause his supporters to deny the legitimacy of the results if he loses.

Last week, Mr. Trump called the presidential election “one big fix” and “one big, ugly lie.”

Jon A. Husted, the secretary of state of Ohio, said it was “wrong and engaging in irresponsible rhetoric” for any candidate to question the integrity of elections without evidence. Mr. Husted, a Republican, said he would have no reason to hesitate to certify the results of the election.

“We have made it easy to vote and hard to cheat,” Mr. Husted said Sunday in an interview. “We are going to run a good, clean election in Ohio, like we always do.”

American elections are, unlike those in many democracies, largely decentralized, rendering the possibility of large-scale fraud extraordinarily unlikely. Further, the balloting in many of the hardest-fought states will be overseen by Republican officials, individuals who would be highly unlikely to consent to helping Mrs. Clinton rig the vote.

Chris Ashby, a Republican election lawyer, said Mr. Trump’s attacks on the electoral process were unprecedented and risked creating a fiasco on Election Day. Mr. Ashby also said that Mr. Trump was “destabilizing” the election by encouraging his supporters to deputize themselves as amateur poll monitors, outside the bounds of the law.