Mr. Bevin, who entered the race as one of the most unpopular governors in the country, saw his approval tank after a face-off with teachers and a push to curb a Medicaid expansion. Many voters soured over a personal style that they saw as overly abrasive and disrespectful.

In turn, Mr. Beshear seized upon that weakness, putting teachers at the center of his campaign, including selecting a high school assistant principal, Jacqueline Coleman, to join his ticket as the candidate for lieutenant governor.

The election had an unusually high turnout of 42 percent, with more than 1.4 million ballots cast — significantly above that of recent governor’s races.

Mr. Beshear found success in cities and in suburbs and in some rural areas as well. He ran up big margins in Lexington and Louisville; won a few historically Democratic turned pro-Trump counties in coal country; picked off suburbs south of Cincinnati that had gone for Mr. Trump; and kept his losing margins largely manageable in conservative rural counties. And while Mr. Bevin won nearly 200,000 more votes than he had in 2015, it was not enough. Mr. Bevin lagged behind the benchmarks he needed to hit in counties across the state.

On Tuesday night, Mr. Beshear claimed victory, but Mr. Bevin resisted conceding and instead began sounding alarms over electoral shortcomings.

In his news conference on Wednesday, Mr. Bevin said “there really are a number of significant irregularities,” but he did not explain, saying his campaign was gathering information that would be “forthcoming in the days ahead.”

“There’s more than a little bit of history of voter fraud in our state,” Mr. Bevin said.

The next step in challenging the results would take place a week from Thursday with a recanvass during which counties would rerun the tabulations on voting machines but not re-examine ballots as would be done in a recount. Mr. Bevin, as a candidate, has the right to seek a recanvassing on any grounds. Kentucky does not have recounts of elections for governor, state election law experts said.