Still, Ms. Sadler did not leave in light of her comments, according to two people familiar with the situation.

Instead, they suggested that Ms. Sadler was pushed out over reports that she had told Mr. Trump that Mercedes Schlapp, the White House strategic communications director, had been the one leaking to the news media. Tensions between the two had reached a point where Ms. Sadler, who worked in the White House communications office focusing on immigration, and Ms. Schlapp were unable to be in the same room together, White House aides said.

The White House in recent weeks has reduced the size of the daily communications meeting, and aides have discussed decreasing the size of the communications staff. Such changes have been threatened before, but on Tuesday, one senior White House official, speaking on the condition of anonymity to discuss internal deliberations, said that more departures may be coming.

In the days after Ms. Sadler’s comments on Mr. McCain, White House officials dodged questions on whether what Ms. Sadler had said was inappropriate. Instead, the press secretary, Sarah Huckabee Sanders, and her deputy Raj Shah have suggested that the leaks were worse than the actual comments. Mr. Shah stressed that the situation was an internal matter, and in an appearance last month on “Fox & Friends,” the president’s preferred TV program, Ms. Sanders targeted leakers in particular.

“I think it is disgusting and some of the most shameful behavior that you can ever engage in,” Ms. Sanders said. “It’s an honor and a privilege to work for the president and to be part of his administration. Anybody who betrays that, I think, is a total and complete coward. And they should be fired.”