Haiti had sent 15,000 people. They “all have AIDS,” he grumbled, according to one person who attended the meeting and another person who was briefed about it by a different person who was there.

Forty thousand had come from Nigeria, Mr. Trump added. Once they had seen the United States, they would never “go back to their huts” in Africa, recalled the two officials, who asked for anonymity to discuss a sensitive conversation in the Oval Office.