The city’s letter on Friday to top officials at the Centers for Disease Control and Prevention suggested that the limited number of tests was already undermining the city’s efforts, citing “slow federal action.”

Alex M. Azar II, the secretary of the Department of Health and Human Services, said on Friday that the C.D.C. had sent enough tests to public health labs across the country for 75,000 people, and that efforts were underway to help the “private sector and hospitals” start testing for the virus.

A spokeswoman for the Association of Public Health Laboratories, Michelle Forman, said there were about 72 public health laboratories that are presently testing for the new coronavirus. “We are not aware of any widespread testing shortages,” she said.

Americans have struggled to make sense of conflicting information from official authorities, including President Trump and members of his own cabinet. Vice President Mike Pence, who previously vowed that “any American could be tested,” conceded on Thursday that “we don’t have enough tests today to meet what we anticipate will be the demand going forward.”

The lack of testing around the country is affecting nursing homes in an unexpected way.

An executive with the American Health Care Association, a trade group representing most of the nation’s 15,700 nursing homes, warned that staff members were far more likely to use protective gear with patients showing any sign of respiratory illness — even as the public is buying masks and the supply chain from China has dwindled.

Nursing homes “everywhere” around the country had begun complaining about shortages of masks and gowns, the executive, David Gifford, said.