Hillary Clinton and Mr. Trump are tied, 41 percent to 41 percent, among likely voters in a three-way race in North Carolina, according to a New York Times Upshot/Siena College poll released on Thursday. Mrs. Clinton leads by two percentage points in a head-to-head contest, 45 percent to 43 percent.

The presidential contest might be the least of the Republican Party’s worries in this rapidly changing state. The embattled Republican governor, Pat McCrory, trails by eight points against Attorney General Roy Cooper, 50 percent to 42 percent.

And even Senator Richard Burr, who was not thought to be in great jeopardy just a few months ago, trails his Democratic challenger, Deborah Ross, by four points, 46-42. That contest is among the handful that seem likely to decide control of the Senate.