Mr. McDaniel and his supporters are exploring legal options to contest Mr. Cochran’s victory, zeroing in on the Cochran campaign’s efforts to bring blacks and other Democratic voters to the polls on Tuesday.

The McDaniel camp charged that a partial review of the tallies in Hinds County had turned up nearly 1,000 ballots that were cast by voters who participated in the Democratic primary on June 3 and were ineligible under state law to vote in the Republican runoff. McDaniel aides said supporters were reviewing ballots across the state, although they have met resistance in about half of the counties they have approached.

Pete Perry, the Hinds County Republican Party chairman, said the McDaniel campaign’s claims were “wildly exaggerated.” In the Jackson precinct at Fondren Presbyterian Church, he said, the McDaniel campaign charged that 192 illegal votes had been cast by people who voted in the Democratic primary. But, he said, only 37 Democrats voted there on June 3.

“Instead of making wild accusations which stir up social media with cries of fraud and corruption, it would be much better for all involved — the voters, the candidates, the 500 poll workers in Hinds County — if we let facts enter into the conversation,” Mr. Perry said.

An aide said the search for irregularities would continue until Mr. McDaniel had the facts to decide whether to contest the results in court. Mississippi law does not allow for a recount, but a judge could order a new vote. And one of the outside conservative groups that bankrolled Mr. McDaniel’s campaign, the Madison Project, said it was standing by the candidate.

“Dirty political tricks are one thing, but if irregular and illegal voting events took place, that is a completely separate matter, and the people of Mississippi deserve straight answers,” the group’s leader, Drew Ryun, said Friday.