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…

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.