Mr. Paul, 57, Republican of Kentucky and a former practicing eye doctor, was “asymptomatic,” his staff said, relieving immediate concerns about his health despite a 2017 injury that affected his lungs. But his diagnosis raised the likelihood that other members of the 100-person Senate, where nearly half are 65 or older, had been exposed and might fall ill or need to self-quarantine just as they were racing to enact the rescue plan.

Mr. Paul had been informed last weekend that he had attended a fund-raiser in Kentucky with two individuals who later tested positive for Covid-19, a warning that sent a handful of other prominent elected officials into self-quarantine out of caution.

Not Mr. Paul.

He pressed on with his Washington schedule anyway. On Wednesday afternoon, he spoke and voted on the Senate floor. He dined at the Republicans’ closed luncheon on Thursday and on Friday, sitting with fellow senators. And after being informed of the diagnosis on Sunday, Senator Jerry Moran, Republican of Kansas, rose to tell fellow Republicans at lunch that he had seen Mr. Paul that morning at the senators-only gym, according to two people who heard him and Mr. Moran’s spokesman.

“I’ve never commented about a fellow Senator’s choices/actions. Never once,” Senator Kyrsten Sinema, Democrat of Arizona, who is known to frequent the senators-only pool, wrote on Twitter. “This, America, is absolutely irresponsible. You cannot be near other people while waiting for coronavirus test results. It endangers others & likely increases the spread of the virus.” (A spokesman for Ms. Sinema said she had been practicing social distancing and had not been to the pool in two weeks.)

The ripples could quickly be felt around the Capitol, one of the few places where work has continued in defiance of public health guidance that has shuttered most workplaces and discouraged large gatherings.