Mr. Trump has repeatedly pointed back to those travel limits to defend his handling of the pandemic, but experts have said the limits were useful mainly to buy time that the administration did not then use to ramp up widespread testing and impose social distancing policies before infections could begin growing exponentially.

By the third week of February, advisers had drafted a list of measures they believed would soon be necessary, like school closures, sports and concert cancellations and stay-at-home orders, but the president did not embrace them until mid-March.

Dr. Fauci, the director of the National Institute of Allergy and Infectious Diseases since 1984, said on Sunday that earlier imposition of such policies would have made a difference.

“I mean, obviously, you could logically say that if you had a process that was ongoing and you started mitigation earlier, you could have saved lives,” he said on “State of the Union” on CNN. “Obviously, no one is going to deny that. But what goes into those kinds of decisions is complicated. But you’re right. Obviously, if we had, right from the very beginning, shut everything down, it may have been a little bit different. But there was a lot of pushback about shutting things down.”

Dr. Fauci’s comments, and the president’s pushback, come at a critical time as Mr. Trump wrestles with how fast to begin reopening the country. Public health experts like Dr. Fauci have urged caution about resuming normal life too soon for fear of instigating another wave of illness and death, while the president’s economic advisers and others are anxious to restart businesses at a time when more than 16 million Americans have been put out of work.

Dr. Fauci and the president have publicly disagreed on several issues, including how long it will take to develop a vaccine and the president’s aggressive promotion of the anti-malaria drug hydroxychloroquine, whose effects are unproven against the coronavirus. At a coronavirus task force briefing last week, Mr. Trump stopped Dr. Fauci from answering a question on the drug.

Dr. Fauci has become a celebrated figure among much of the public, which trusts him far more than Mr. Trump, according to polls. A Quinnipiac University survey last week found that 78 percent of Americans approved of Dr. Fauci’s handling of the crisis compared with 46 percent who approved of the president’s response. That has prompted resentment among other government officials, some of whom have privately criticized Dr. Fauci for playing to the media and not always sending consistent messages.