In interviews, people who have seen Mr. Cruz speak privately said he was able to switch back and forth with ease between his crusading public persona and a softer and less confrontational tone.

Mr. Perkins, the evangelical leader, who endorsed Mr. Cruz this week, said that there was nothing unusual about this. “You find a point of commonality,” he said. “That doesn’t mean you’re not for the other issues.” Mr. Cruz, he added, was sincere in his beliefs. “I’ve never seen him speak out of both sides of his mouth.”

Robert O’Brien, a Los Angeles lawyer who supported Mitt Romney in 2012 and Gov. Scott Walker of Wisconsin before he dropped out, gathered more than a dozen former Romney supporters to meet Mr. Cruz in December 2014. “I found his positions to be solid conservative Ronald Reagan-type positions,” said Mr. O’Brien, saying Mr. Cruz, whom he found charming, focused primarily on national security. “I didn’t find them to be kind of the caricature that came out of him in the Senate,” he said. But he said he did not hear any contradictions with Mr. Cruz’s public pronouncements.

Despite his public denunciations of political opponents and firebrand views on social issues, Mr. Cruz has a long history with some of the party’s prominent donors who hold more liberal views on social issues.