Mr. Trump’s last-minute rally for Mr. Bevin in Lexington on Monday night, which filled the usually Kentucky Wildcat-blue Rupp Arena with Louisville Cardinal-red “Make America Great Again” hats, could be particularly important, said Al Cross , a Kentucky political commentator and director of the Institute for Rural Journalism and Community Issues .

Mr. Cross noted that Mr. Bevin, a brash investment manager and former Army officer who is seeking a second term, is not particularly popular — one poll ranks him the second most unpopular governor in the nation. But Mr. Trump still is a galvanizing presence for his supporters. And with few voters clued in to an off-year election, Mr. Cross said, “It’s important to have that final push from el jefe.”

Mr. Bevin, 52, became a first-time political officeholder after winning a 2015 campaign in which he marketed himself as an outsider intent on bringing business-friendly values to the state. The victory was an important one for Republicans, who have dominated presidential elections in Kentucky elections in recent years, but whose success in governor’s races lagged.

As governor, Mr. Bevin has seemed to relish attacking those who have criticized or pushed back on his agenda, Democrats and Republicans alike — and, perhaps most significantly, public-school teachers. When educators walked off the job last year to oppose budget cuts and proposed changes to teacher pensions, Mr. Bevin called protesters “selfish” and “ignorant,” blaming them for the shooting of a young girl and accusing them of creating the opportunity for child sexual assaults by leaving children unattended at home.