“The question is, is this a kind of local outbreak, or is this the first signature of a much larger outbreak?” said Aubree Gordon, a professor of epidemiology at the University of Michigan’s School of Public Health. “I think it’s going to be very important that local public health officials and the government react quickly and appropriately to try to stop the spread.”

Since the nation’s first coronavirus case was found in Washington six weeks ago, health officials have conducted sporadic testing, limited by narrow guidelines from the Centers for Disease Control and Prevention for who was at risk, as well as difficulties with testing kits.

On Monday, the state increased its testing and was now handling about 200 requests each day, said Dr. Kathy Lofy, the health officer of Washington State. Much of the focus has been in the Seattle area, but Dr. Lofy said that could change. “Given the movement of people around the state, it is possible the virus is spreading in other counties, too,” she said.

Dow Constantine, the executive of King County, where officials were trying to buy a motel as a possible isolation facility, said it could be open as early as the end of the week. He said he also hoped to repurpose modular housing units, initially destined for the region’s homeless crisis, to be used to help isolate people and free up space at medical centers.

“We need that hospital capacity for treatment,” Mr. Constantine said.