The outbreak infected a third Republican senator on Saturday as Ron Johnson of Wisconsin reported testing positive as did former Gov. Chris Christie of New Jersey, who helped prepare Mr. Trump for his campaign debate on Tuesday. Other Republicans close to Mr. Trump were being tested and awaiting results as the weekend opened. In the course of barely 24 hours, the president, his wife, his campaign manager, his party chairwoman, his senior adviser, his former counselor and now three Republican senators have all tested positive for the virus, along with several reporters.

The White House medical unit was investigating Mr. Trump’s announcement of his nomination of Judge Amy Coney Barrett to the Supreme Court last weekend as a likely source of the virus’s spread. More than a half-dozen people who attended the event on Sept. 26 or were with the president on Air Force One flying to a campaign rally in Pennsylvania later in the evening have now tested positive.

While the ceremony itself was outdoors in the Rose Garden, most of the people in attendance other than reporters were not wearing masks or keeping socially distant. A number of people also joined Mr. Trump and Judge Barrett inside the White House for a reception, again without masks and in some cases shaking hands and hugging without evident regard for the dangers of the virus.

With the election just 31 days away, White House officials sought on Saturday to project as much of a business-as-usual image as possible, insisting that the president can govern the country from his hospital bed and that there was no need to transfer power to Vice President Mike Pence. Even as doctors hovered over Mr. Trump, his staff on Friday night issued a report on his buy-American drive and announced some minor appointments. On Saturday morning, the White House announced that the president had signed two bills appointing members to the Board of Regents of the Smithsonian Institution.

Mr. Trump has been suffering from a cough, congestion, fatigue and a fever, according to people informed about his condition, and some of the symptoms worsened as Friday progressed including the drop in oxygen level that alarmed the president. Plans for him to convalesce at the White House were abruptly scrubbed in favor of sending him to the hospital for what officials said would be a few days.

One reason, according to an administration official, was that it would be better for the president to leave while he could still walk on his own power to the helicopter rather than risk getting sicker and having the politically damaging image of needing assistance to be transported to the hospital later.

At Walter Reed, doctors put Mr. Trump on remdesivir, an antiviral drug that has hastened the recovery of some coronavirus patients, which came after a variety of other treatments were administered while the president was still at the White House. Mr. Trump received a single 8-gram dose of polyclonal antibody cocktail, an experimental treatment that the White House obtained special permission to try, as well as zinc, vitamin D, an anti-heartburn medicine called famotidine, melatonin and aspirin.

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