SALT LAKE CITY — Donald J. Trump has vowed to turn blue states red in November, but as he continues to rattle Republicans with his combative style and provocative proposals, voters in some of the country’s most conservative strongholds are considering a radical idea: supporting Hillary Clinton.

The dilemma posed by this year’s choice of candidates is perhaps most apparent here in Utah, a mountain state that has not backed a Democrat for president since Lyndon B. Johnson in 1964. More than 50 years later, a large Mormon population with a strong distaste for Mr. Trump has left the state up for grabs, and with a substantial Mormon presence spilling into places such as Arizona, Idaho and Nevada, what would normally be a Republican safe zone could be surprisingly competitive.

With the Clinton campaign looking to put Republican-leaning states in play, the decision for many Mormon voters in Utah has become agonizing as they digest Mr. Trump’s stances toward Muslims in light of their own history as an oft-maligned religion, and as his “America First” message repels well-educated Mormons who travel the world on missions and who welcome refugees.

“People who normally vote Republican are in a terrible state of ambivalence right now,” said Tim Chambless, a political scientist at the University of Utah. “They are so undecided. They want to vote, but they aren’t sure how to vote.”