Democrats are also taking Utah seriously. While they realize that their improved chances in the state are not because their nominee has suddenly surged in popularity, the party would be happy to break its losing streak.

“This is the first time since the mid-1960s that a Democratic presidential candidate could win in Utah,” said Peter Corroon, the party’s chairman in the state. “Unfortunately, it’s not because of the Democrat, it’s because of the Republican.”

Nonetheless, the Clinton campaign has staff on the ground in Utah, and it is dispatching former President Bill Clinton to the state for a fund-raising event this week. The campaign would not say if Mrs. Clinton would make a trip of her own, but the possibility remains.

“There is no doubt that Donald Trump’s offensive rhetoric has made Utah more competitive than before, and we will continue to assess our options in the state,” said Marlon Marshall, the Clinton campaign’s director of state campaigns and political engagement.