FORT THOMAS, Ky.—Some Republicans in this Cincinnati suburb of tidy homes and tree-lined streets, long a GOP stronghold, say they are unhappy with President Trump’s tone and behavior.

Their unease is a potential warning sign for the GOP less than a year ahead of the 2020 presidential election.

The 16,000-person city is overwhelmingly white and home to a lot of well-off, college-educated people. Its demographics are similar to suburbs where some Republican candidates recently struggled, including those outside Philadelphia, Charlotte, N.C., and Richmond, Va.

Few doubt the president will capture the area next year. In 2016, he carried Campbell County, which encompasses Fort Thomas, by 24 percentage points.

But Sarah Cameron, chairwoman of the Campbell County GOP and a Fort Thomas resident, said she thought enthusiasm for the president was dampening among some Republicans who supported him four years ago.