During his cabinet meeting on Wednesday, President Trump casually mentioned that Lee Zeldin, the administrator of the Environmental Protection Agency, intended to fire 65 percent of employees, an incision so deep that officials said it would hobble the E.P.A.

Mr. Trump said Mr. Zeldin “thinks he’s going to be cutting 65 or so percent of the people from environmental. And we’re going to speed up the process, too, at the same time.”

Within minutes, managers at the agency said they received a White House memo telling them to prepare for mass layoffs.

The memo, which was sent to leaders of multiple agencies, said that the federal government “is costly, inefficient, and deeply in debt.” It did not mention the 65 percent goal, but laid out steps for the E.P.A. to prepare for what is known as a reduction in force, which would result in eliminating jobs.

The agency did not respond to a request to discuss Mr. Zeldin’s job-cutting goal.

The E.P.A. had 15,123 full-time employees at the end of December, according to the latest budget. A reduction of 65 percent would mean the loss of nearly 10,000 jobs, which would devastate the agency responsible for clean air and clean water, said Marie Owens Powell, the president of the agency’s biggest union, the American Federation of Government Employees.

Ms. Owens Powell said the administration had not informed the union about its 65 percent goal and that she had heard about it first from Mr. Trump on television.