“I would hope that doctors would stick to the science and try to keep a cool head,” said Dr. Percio S. Gulko, chief of the division of rheumatology at the Icahn School of Medicine at Mount Sinai. “Somebody is prescribing it for people who are trying to get it, in some instances preventively. They may just be depriving the patients who do need it for an established indication, for a possibility or a speculation.”

“If it does turn out to be a success, we understand that there will be a need for more than has ever been available for patients with autoimmune diseases,” said Dr. David R. Karp, the president-elect of the American College of Rheumatology and chief of the rheumatic disease division at the University of Texas Southwestern Medical Center in Dallas. “We hope there will be a way for our patients to continue to access these medicines they’ve been taking for many years. Other medications can be used, but the safety profile gets much worse, and patients will likely have side effects.”

Judie Stein, of Sun Prairie, Wis., said she was stunned when she heard Mr. Trump pronounce the name of the drug she has taken for two years to treat rheumatoid arthritis. “When I was first prescribed this, nobody had heard of it,” said Ms. Stein, who is 59. She said she has a one-month supply of the drug and she quickly tried to refill her prescription yesterday. She has not yet received a confirmation. “If it’s readily available and has other uses, then fine, but when I say I really need it, I need it,” she said.

Dr. Michael Belmont, medical director of NYU Langone Orthopedic Hospital, said a number of his lupus patients had requested 90-day prescriptions for hydroxychloroquine, rather than the usual 30 days’ worth.

Noting that hydroxychloroquine was being widely used in coronavirus patients outside of controlled studies, he said, “It would be a shame if we use a lot of this and after all is said and done we are not able to determine with accuracy whether it had an effect or not.”

Onisis Stefas, the chief pharmacy officer for Northwell Health’s 23 hospitals, said the system began stocking up on hydroxychloroquine several weeks ago. He said the drug was being given to many coronavirus patients, but that Northwell’s 10 pharmacies were also setting aside supplies for patients who had been taking it regularly for lupus or rheumatoid arthritis.

Concerned about shortages, Mr. Stefas said, “The last thing I want to happen now is that, especially since President Trump and others have been mentioning this by name, is that people will go out and ask their doctors to write prescriptions, just in case.”