The agency leading the nation’s coronavirus response said seven employees had tested positive for the virus, with another four cases pending, though in a letter to its employees’ union, it declined a request to say where they were located, prompting criticism from the union that the agency was jeopardizing public health.

Union leaders last week had asked the agency, Federal Emergency Management Agency, how many employees had tested positive, and in which offices, so that workers who might have interacted with those people could decide whether to get tested as well. On Friday, FEMA turned down the request, saying the union did not need to know, according to a copy of the agency’s letter that was reviewed by The New York Times.

In response to inquiries from The Times, the agency on Saturday said that seven employees had tested positive for the coronavirus. “Currently, FEMA has 11 total cases — seven employees have tested positive and four potential cases are pending,” Lizzie Litzow, a spokeswoman for FEMA, said in a statement. “Individuals who need to be aware of their names and locations have been made aware.”

Steve Reaves, president of the American Federation of Government Employees Local 4060, which represents about 5,000 FEMA employees, said that by not sharing details about the staff infections with the union, the agency was endangering other employees as well as the safety of the people to whom the agency was currently providing aid. Over all, the agency has about 14,000 employees.

“If we’re out there handing out masks and gloves, and we’ve got Covid, then they’re contaminated,” said Mr. Reaves, referring to the disease caused by the coronavirus.

The concern over the safety of FEMA employees comes as the agency is already stretched thin by three years of major natural disasters, and as it has been forced to rapidly rethink its recommendations for emergency relief shelters, which often accommodate many survivors in close quarters, risking transmission of disease.