A series of New York Times/Siena College polls of swing states conducted last month found that in the rural Appalachian counties of Pennsylvania — the vast majority of which Mr. Trump won in 2016 — only 44 percent of voters said they would definitely choose him over a Democratic candidate in 2020.

By comparison, Mr. Trump commanded the backing of well over 60 percent of voters in most of those counties in 2016 — reaching as high as 80 percent support in some of them.

In Kentucky, where the Democratic challenger, Andy Beshear, leads in the race for governor — though the Republican incumbent, Gov. Matt Bevin, has refused to concede — it was in the rural, Appalachian counties across the eastern part of the state where Mr. Bevin had the most trouble matching the level of support Mr. Trump had in 2016: Just 60 percent of voters in those counties went for Mr. Bevin, compared with the 78 percent who voted for Mr. Trump three years ago.

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Much of this decline stems from Mr. Bevin’s personal unpopularity: His approval rating has remained under 50 percent since 2017, according to Mason-Dixon polling, partly driven by his high-profile attacks on public-school teachers and efforts to restrict access to Medicaid. Polls show that he is less liked than Mr. Trump throughout the state.

Still, there are warning signs for Mr. Trump within Mr. Bevin’s struggles. A Mason-Dixon poll conducted just before this week’s election found that in most parts of Kentucky, Mr. Trump’s favorability rating was just a few points lower than his share of the vote in 2016. But in those eastern, Appalachian counties, the president’s 67 percent favorability rating — while still high — sits 11 points below his 2016 vote share there. His support appears to have fallen more in rural eastern Kentucky than anywhere else in the state.