Frank Wucinski and his 3-year-old daughter, Annabel, are among the dozens of Americans the government has flown back to the country from Wuhan, China, and put under quarantine to check for signs of coronavirus.

Now they are among what could become a growing number of families hit with surprise medical bills related to government-mandated actions.

Mr. Wucinski, a Pennsylvania native who has lived in China for years, accepted the U.S. government’s offer to evacuate from Wuhan with Annabel in early February as the new coronavirus spread. His wife, who is not an American citizen and remains in China, developed pneumonia that doctors think resulted from Covid-19, the disease caused by the respiratory virus. Her father, whom she helped care for, was infected and recently died.

The first stop for Mr. Wucinski and Annabel was a two-week quarantine at Marine Corps Station Miramar near San Diego. During that time, they had two mandatory stays in an isolation unit at a nearby children’s hospital. The first started upon arrival in the United States, and the second was a few days later, after an official heard Annabel coughing.

“The hospital staff were very nice, they brought us a lot of toys,” Mr. Wucinski said. “Each time it was three or four days. I love my daughter to death, but being in the same room that long, she is not a great conversationalist.”

Both have repeatedly tested negative for the virus.

After their release from quarantine, Mr. Wucinski and his daughter went to stay with his mother in Harrisburg, Pa. That’s where they found a pile of medical bills waiting: $ 3,918 in charges from hospital doctors, radiologists and an ambulance company.

“I assumed it was all being paid for,” Mr. Wucinski said. “We didn’t have a choice. When the bills showed up, it was just a pit in my stomach, like, ‘How do I pay for this?’”

Mr. Wucinski’s employer, a standardized testing company, provided health benefits when he lived in China but does not offer coverage in the United States.

Patients in the United States regularly confront surprise medical bills that are hard to decode. Mr. Wucinski’s case suggests that those held in mandatory isolation for suspected coronavirus may be no exception.

The federal government has the authority to quarantine and isolate patients if officials believe them to be a public health threat. These powers, which date back to cholera outbreaks among ship passengers in the late 19th century, are rarely used. They don’t say anything about who pays when the isolation happens in a nongovernmental medical facility — or when they’re brought there by a private ambulance company.

“There is no uniform practice,” said Lawrence Gostin, a professor of global health law at Georgetown University. “They do have the powers, but they’ve almost never used them in modern times.”

Few patients have been held in mandatory isolation, but the number is likely to grow if the coronavirus continues to spread across the United States. Eleven cases were confirmed in San Antonio on Friday evening. Earlier in the day, public health authorities identified a second case in California and a first in Oregon in which patients who had not traveled to an affected country became infected.

A Centers for Disease Control and Prevention spokesman declined to comment on whether it would pay the bills of patients kept in mandatory isolation.

Mr. Gostin worries that high charges for mandatory isolation could make patients wary of seeking needed medical treatment.

“The most important rule of public health is to gain the cooperation of the population,” he said. “There are legal, moral and public health reasons not to charge the patients.”

These hospital stays could prove expensive. The International Federation of Health Plans estimates that the average day in a U.S. hospital costs $ 4,293, compared with $ 1,308 in Australia and $ 481 in Spain. The hospital stays may be especially costly for patients without health insurance or for those who have large deductibles, which they must pay before their health benefits kick in.

Mr. Wucinski recalls other patients asking about how medical bills would be handled during daily town hall meetings for those quarantined at the Marine Corps station. He felt the answers weren’t clear.

He did receive a document upon leaving quarantine directing him to contact a government email address with any medical bills. He sent an email on Feb. 24 detailing the charges and asking what would be done.

“My question is why are we being charged for these stays, if they were mandatory and we had no choice in the matter?” Mr. Wucinski wrote in his message.

He has not received a response, he said. When contacted by The New York Times, a Rady Children’s Hospital spokesman said the physicians’ bill had been sent in error and that the family would not be held responsible for the charges.

“We’re in the process of assessing how the error occurred,” the spokesman, Benjamin Metcalf, said. “We are working with government agencies regarding billing for these cases.”

But the hospital bill represented only a fraction of those the family received.

The ambulance company that transported the Wucinskis, American Medical Response, charged the family $ 2,598 for taking them to the hospital. A company representative declined to comment on the bill “due to patient privacy concerns,” but said the company would look into the case.

An additional $ 90 in charges came from radiologists who read the patients’ X-ray scans and do not work for the hospital. Having such doctors, who may be outside a patient’s insurance networks, provide services to hospital patients is one of the major causes of surprise medical bills.

The radiologists’ employer, San Diego Medical Imaging Group, did not respond to a request for comment.

Mr. Wucinski is looking into whether he or his daughter might qualify for Medicaid, the public insurance program that covers low-income Americans. Some news outlets have seized on the fact that he coughed enough on a recent television interview to require water.

Mr. Wucinski recently noticed that his daughter is blinking a lot and hopes to get the issue examined by a doctor, but is worried about the charges they may face. He has wondered whether the stress of the past month and the separation from her mother has played a role.

“I should take her to the doctor this week for a checkup, but we don’t have insurance, so it’s just going to have to be cash,” he said.