Donald J. Trump is heading west this weekend for rallies in Las Vegas and Phoenix and a fund-raiser at Barry Goldwater’s old estate, known as Be-nun-i-kin, Navajo for “house on top of the hill.”

But it was the prospect of Mr. Trump’s gathering with members of the Navajo Nation that was creating the most intrigue around the trip of the presumptive Republican presidential nominee.

Navajo leaders had extended an invitation to Mr. Trump, as they have to other candidates, to meet with them, despite objections from a number of members over his repeated use of the name “Pocahontas” to deride Senator Elizabeth Warren of Massachusetts. On Friday, Navajo officials said the meeting was not going to happen during Mr. Trump’s trip.

Nevertheless, Mr. Trump’s words have stirred a debate among Native Americans about how they should deal with him. For many, his mention of the historical figure is offensive and a sign that Mr. Trump, who has been accused of being anti-immigrant, also has problems with the people who first inhabited the country.