FRANKFORT, Ky. — After coming roughly 5,000 votes short in his re-election campaign for Kentucky governor, Matt Bevin on Wednesday called for votes across the state to be recanvassed as he expressed concerns about unspecified voting irregularities.

“We want the people of Kentucky to have absolute confidence that their votes were counted as they should have been counted,” Mr. Bevin told reporters on Wednesday while standing in the entryway of the governor’s mansion he was fighting to keep.

Experts say a recanvassing is unlikely to swing the outcome of the election. Even so, Mr. Bevin has the potential to set in motion a challenge that could escalate into a showdown in the State Legislature, where lawmakers could ultimately determine the result.

Legal scholars and critics raised concerns about the governor leveling accusations of irregularities without providing evidence. And as Mr. Bevin prepares for a review of the vote, his Democratic challenger, Attorney General Andy Beshear, has plowed ahead with planning his agenda and staff as governor.

During a news conference in Louisville on Wednesday morning, Mr. Beshear stood at a lectern where the logo from his campaign had already been altered to declare him “governor-elect,” and said he was beginning the work of sketching out his first budget proposal in January.

“The politics part of this is over,” he said. “It’s time for governance.”

Mr. Bevin and Mr. Beshear have said they have not spoken to each other. But Mr. Beshear’s campaign has urged the governor to stand down and concede, saying that the governor had virtually no shot at taking away Mr. Beshear’s considerable lead.

“We hope that Matt Bevin honors the results of the recanvass, which will show he received fewer votes than Andy Beshear,” Eric Hyers, Mr. Beshear’s campaign manager, said in a statement issued later on Wednesday.

Mr. Bevin, the Republican incumbent in a conservative state, increasingly came to rely on President Trump, who hosted a rally with Mr. Bevin in a state he won in 2016 by close to 30 percentage points.

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.

Describing past recanvass efforts, Joshua A. Douglas, a professor of law at the University of Kentucky, said, “In every instance, the vote totals change by a very small amount.” He added that it would likely come “nowhere near” the 5,000 votes Mr. Bevin needs to close the gap.

Mr. Bevin described recanvassing as a relatively routine undertaking, largely meant to confirm the integrity of the election process — making sure, as he put it, “there wasn’t a transposed number or something of this sort.”

It is unclear, however, how far he is willing to pursue his challenge.

“I’m confident that in the end the right results will be delivered,” Mr. Bevin said, “and I will be entirely comfortable with whichever way they go if I’m confident the process has been served.”

Professor Douglas said Mr. Bevin’s recourse after a recanvassing would be contesting the election with the State Legislature, which is controlled by Republicans. There, a group of randomly selected lawmakers (eight from the House, and three from the Senate) would form an elections board that would hear evidence and arguments before arriving at a recommendation that would be forwarded to the entire Legislature.

Lawmakers could end up deciding the contest. The last time a governor’s race reached the Legislature was 1899, and it ended with one of the candidates being assassinated.

Campbell Robertson contributed reporting from Louisville, Ky.