“They should lay out their cards so everyone knows fact from fiction and fact from innuendo,” said Dallas Woodhouse, the executive director of the North Carolina Republican Party. “If they can state with certainty that whatever bad behavior that took place could not have changed the race, then they should certify Mr. Harris. If they can state there was a substantial likelihood that the race could have been altered, then we would not oppose a new election.”

The deepening concerns about the Ninth District center on Mr. Harris’s voter turnout operation, which partly relied on L. McCrae Dowless Jr., a political operative from Bladen County with both a felony record for fraud and an extensive history of working for Democratic and Republican campaigns. Witnesses have repeatedly described an effort that involved at least one seemingly illegal tactic: collecting absentee ballots directly from voters. The approach stirred worries that ballots may have been improperly marked for Mr. Harris or discarded if they were to be cast for Mr. McCready.

Mr. Dowless declined to comment, and Mr. Harris and his supporters have denied wrongdoing.

Democrats have warned that if the allegations about Mr. Harris’s campaign are not sufficiently settled, they would be reluctant to allow him to be seated in January. Ms. Pelosi reasserted on Thursday that the House would have the final say on who was seated and whether to investigate the election itself.

“The House still retains the right to decide who is seated,” Ms. Pelosi said. “Any member-elect can object to the seating of, the swearing in of another member-elect, and we’ll see how that goes.”

If North Carolina officials are not able to resolve the allegations in an acceptable way, she continued, the House Administration Committee has the authority to investigate the matter itself and ultimately determine a winner. If it was unable to, she added, the committee could call for a new election. (The House has the constitutional authority to be “the judge of the elections, returns and qualifications of its own members.”)