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President Trump has praised Dr. Anthony S. Fauci as a “major television star.” He has tried to demonstrate that the administration is giving him free rein to speak. And he has deferred to Dr. Fauci’s opinion several times at the coronavirus task force’s televised briefings.

But Dr. Fauci, the director of the National Institute of Allergy and Infectious Diseases since 1984, has grown bolder in correcting the president’s falsehoods and overly rosy statements about the spread of the coronavirus in the past two weeks — and he has become a hero to the president’s critics because of it. And now, Mr. Trump’s patience has started to wear thin.

So has the patience of some White House advisers, who see Dr. Fauci as taking shots at the president in some of his interviews with print reporters while offering extensive praise for Mr. Trump in television interviews with conservative hosts.

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Mr. Trump knows that Dr. Fauci, who has advised every president since Ronald Reagan, is seen as credible with a large section of the public and with journalists, and so he has given the doctor more leeway to contradict him than he has other officials, according to multiple advisers to the president.

When Mr. Trump knows that he has more to gain than to lose by keeping an adviser, he has resisted impulses to fight back against apparent criticism, sometimes for monthslong interludes. One example was when he wanted to fire the White House counsel, Donald F. McGahn II, in 2017 and early 2018. Another was Jeff Sessions, the former attorney general. Mr. Trump eventually fired both when he felt the danger in doing so had passed.

So far, the president appears to be making the same calculation with Dr. Fauci, who was not on the briefing room podium on Monday evening. When asked why, Mr. Trump said he had just been with Dr. Fauci for “a long time” at a task force meeting. Officials, asked about the doctor’s absence, repeated that they were rotating officials who appear at the briefings.

“He’s a good man,” Mr. Trump said. Dr. Fauci was scheduled to be on Fox News with Sean Hannity a short time later.

Still, the president has resisted portraying the virus as the kind of threat described by Dr. Fauci and other public health experts. In his effort to create a positive vision of a future where the virus is less of a danger, critics have accused Mr. Trump of giving false hope.

Dr. Fauci and the president have publicly disagreed on how long it will take for a coronavirus vaccine to become available and whether an anti-malaria drug, chloroquine, could help those with an acute form of the virus. Dr. Fauci has made clear that he does not think the drug necessarily holds the potential that Mr. Trump says it does.

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.”

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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.”

And Dr. Fauci is savvy not just about the inner workings of the government but about the news media that covers it.

When Vice President Mike Pence took over as the lead of the coronavirus task force, his advisers wanted to put a 24-hour pause on interviews that administration officials were giving as they assessed where the administration was after a chaotic few weeks. They were initially fine with Dr. Fauci’s appearances, meeting with him before interviews to get a sense of what he planned to say.

But in the past two weeks, as Dr. Fauci’s interviews have increased in frequency, White House officials have become more concerned that he is criticizing the president.

Officials asked him about the viral moment in the White House briefing room, when he put his hand to his face and appeared to suppress a chuckle after Mr. Trump referred to the State Department as the “Deep State Department.” Dr. Fauci had a benign explanation: He had a scratchy throat and a lozenge he had in his mouth had gotten stuck in his throat, which he tried to mask from reporters.

Some officials have not questioned that Dr. Fauci is giving interviews, but they have wondered how he has so much time for so many requests from the news media.

Dr. Fauci, for his part, has been dismissive of some questions about whether he was at odds with the president, treating it as a news media obsession.

“I think there’s this issue of trying to separate the two of us,” he said on CBS.