Dr. Agus has boasted to senior administration officials that he has talked about the benefits of the drugs with Mr. Trump, who has publicly counseled that medical efforts not be wedded to traditional drug trials. That has alarmed senior health officials, who see the Oracle project as a way around the rigor and consensus of the government’s drug safety oversight.

On Tuesday afternoon, Dr. Fauci met with Drs. Deborah L. Birx, the White House’s coronavirus coordinator, Robert R. Redfield, director of the Centers for Disease Control and Prevention, and Stephen M. Hahn, the F.D.A. commissioner, to go over their concerns with the Oracle project and review new Chinese data that indicated the drugs have no meaningful effect.

One person familiar with the Oracle discussions said the project would be driven by science and data, not political calculations. Another person familiar with the conversations said there had been other talks between the Trump administration and television medical experts, such as Dr. Mehmet Oz, a widely watched and often criticized television personality who joined a Fox News-hosted virtual town hall with the president and Vice President Mike Pence on Tuesday to ask a question about hydroxychloroquine.

In response to the question, Mr. Pence said that patients should defer to their doctors on whether to take the drug.

On some of the president’s favorite Fox News programs in recent weeks, including Laura Ingraham’s, guests have pitched chloroquine as a coronavirus treatment. On March 14, Dr. Marc K. Siegel, a Fox News medical correspondent, said on “Justice with Judge Jeanine” that “no one is talking about" antiviral treatments, but that “we can try some of these treatments.”

Mr. Trump’s own chloroquine boosterism began as early as last Wednesday in a private Oval Office meeting, where he told a group of top officials, including Dr. Hahn, that promoting the drug would be a shot of inspiration for the public. The F.D.A., Mr. Trump said in the meeting, should promote chloroquine treatment, two senior administration officials said.

Early that morning, and to the surprise of top officials at the F.D.A., Mr. Trump wrote on Twitter that he would be holding a news conference “to discuss very important news from the F.D.A. concerning the Chinese Virus!” Pleading with the White House, officials at the F.D.A. were able to hold it off, according to a senior administration official, forcing Mr. Trump to take his message to the next day’s coronavirus task force news briefing, where he told reporters that chloroquine would be distributed to “large groups of people” even before the government had concluded studying its safety and effectiveness.