In an interview with Science Magazine, Dr. Fauci responded to a question about how he had managed to not get fired by saying that, to Mr. Trump’s “credit, even though we disagree on some things, he listens. He goes his own way. He has his own style. But on substantive issues, he does listen to what I say.”

But Dr. Fauci also said there was a limit to what he could do when Mr. Trump made false statements, as he often does during the briefings.

“I can’t jump in front of the microphone and push him down,” Dr. Fauci said. “OK, he said it. Let’s try and get it corrected for the next time.”

In an interview with CBS’s “Face the Nation” on Sunday, Dr. Fauci played down the idea that there was a divide between him and the president. “There isn’t fundamentally a difference there,” he said.

“The president has heard, as we all have heard, what are what I call anecdotal reports that certain drugs work. So what he was trying to do and express was the hope that if they might work, let’s try and push their usage,” Dr. Fauci said. “I, on the other side, have said I’m not disagreeing with the fact anecdotally they might work, but my job is to prove definitively from a scientific standpoint that they do work. So I was taking a purely medical, scientific standpoint, and the president was trying to bring hope to the people.”

A White House spokesman and Dr. Fauci did not respond to requests for comment.

Dr. Fauci came to his current role as the AIDS epidemic was exploding and President Reagan was paying it little attention. He and C. Everett Koop, the surgeon general, were widely credited with spurring the Reagan administration to action against AIDS, a fact that underscores Dr. Fauci’s ability to negotiate difficult politics.

He has recognized Mr. Trump’s need for praise; in the president’s presence and with audiences that are friendly to him, Dr. Fauci has been complimentary. He told the radio host Mark Levin on Fox News of the administration’s response to the virus: “I can’t imagine that under any circumstances that anybody could be doing more.”