In private, Mr. Trump has made fun of his adviser’s militant reputation, suggesting that he was the one restraining Mr. Bolton rather than the other way around. “If it was up to John, we’d be in four wars now,” one senior official has recalled the president saying.

At a recent meeting of the Republican Jewish Coalition, Mr. Trump pulled Mr. Adelson aside and asked how he thought Mr. Bolton was doing, according to a person briefed on the conversation. Mr. Adelson said that if Mr. Trump was happy, then he was happy.

But it was not clear if he was happy. When Mr. Trump sours on an adviser, he often conducts informal surveys of people in his circle asking how they think that adviser is doing — it is his way of contemplating whether he should make a change. In the case of Mr. Adelson, the president may have also been testing the reaction of Mr. Bolton’s backer, wary of offending the Republican megadonor.