Mr. Harris has denied wrongdoing, and party officials have not publicly backed away from their support of his candidacy. But Republican support for the bill seems to reflect the increasing sense on the part of both parties that a new election is becoming increasingly likely.

The legislation approved Wednesday, first by the State House and soon after by the State Senate, creates the possibility that Robert M. Pittenger, the incumbent, could again face off against Mr. Harris in a rematch of the Republican primary that Mr. Harris won with the help of a significant number of absentee votes. It was not immediately clear whether Mr. Pittenger would mount another campaign if the state board called a new election.

Beyond Mr. Pittenger, some Republicans had privately speculated that former Gov. Pat McCrory, who lives in the district, could run for the seat if there was a new primary. But Mr. McCrory said in an interview on Tuesday that he had “no plans to run for that office.”

On Wednesday, Ford Porter, spokesman for Gov. Roy Cooper, did not say whether the Democratic governor would sign the legislation that could trigger a new primary, but released a statement that said the state’s residents “deserve honest and fair elections and the governor is reviewing this legislation carefully.”

Mr. Dowless, who has declined to comment, has a criminal history that includes felony convictions for fraud and perjury. He has been named as a “person of interest” in the state board’s investigation into whether he or people working for him illegally handled or manipulated absentee ballots.