Ted Cruz sounded despondent at the possibility that same-sex marriage could become legal when he called into the radio show of Tony Perkins, a vehement opponent of gay rights, in February 2014. “Our heart weeps for the damage to traditional marriage that has been done,” Mr. Cruz said, urging conservatives to pray but also to fight back. “Be as wise as serpents and gentle as doves,” he said, quoting Scripture.

But seven months later, when Mr. Cruz visited the Midtown Manhattan office of a major Republican supporter of same-sex marriage, he sounded almost indifferent: If New York politicians wanted to legalize it, that was their business, he told Paul Singer, the billionaire investor who had bankrolled efforts to strike down laws forbidding same-sex marriage across the country. There was no reason the issue had to drive a wedge between the two men, the Texas senator said, according to a person briefed on the meeting who supports one of Mr. Cruz’s opponents. (Mr. Singer went on to endorse Senator Marco Rubio of Florida a year later.)

Mr. Cruz would hardly be the only person to modulate his most provocative language when speaking to people whose support he is courting. But Mr. Cruz, the winner of the Iowa caucuses, has worked harder than any Republican running for president to portray himself as the true and pure conservative and to impugn his rivals as not faithful enough in their conservatism.

Now, as he finds himself locked in an intense struggle for the support of the party’s top donors, instances in which he curried favor, or merely associated, with people who do not share his views on some contentious issues are being dredged up and shared by people who believe that Mr. Cruz has been disingenuous, especially in his solicitation of money from people he has coolly dismissed as having “New York values.”