Ms. Brennan told lawmakers on Thursday that the agency was already in talks with the Treasury about the potential loan, but its revenue predictions suggest that the money would not be enough if the crisis continues.

Even with an increase in online shopping and package delivery to Americans cooped up at home, the agency could see a 50 percent reduction in total mail volume by the end of June, compared with the same period last year, Ms. Brennan told the lawmakers.

She said the projected shortfall this fall could throw regular mail delivery into doubt.

“They are chilling numbers,” said Representative Gerald E. Connolly, Democrat of Virginia, who leads the House subcommittee responsible for the Postal Service. “The reaction of a lot of my colleagues — their jaws were dropping. It is one thing to say the Postal Service is suffering. It is another to hear these specifics.”

For now, the mail service, which operates under government-mandated service requirements, has continued uninterrupted during the pandemic. Even as scores of its more than 600,000-person work force have fallen ill and some have died, mail sorters and carriers have continued to walk their routes in every corner of the country, in many cases the only physical lifeline Americans now have to the outside world. They deliver medicines, coronavirus test kits and packages ordered online, and could play a crucial role in November’s presidential election, in which voting by mail is expected to surge.

But the debate over whether to shore up the Postal Service has been politically fraught.

Negotiators on Capitol Hill had reached a tentative deal last month to provide the Postal Service around $13 billion in direct relief as part of the $2 trillion stimulus law. That was far less than House Democrats had proposed, but it had the buy-in of a crucial Republican negotiator: Senator Ron Johnson of Wisconsin, the chairman of the Senate Homeland Security and Governmental Affairs Committee, according to the officials familiar with the talks.

But Mr. Mnuchin said the administration would not have it.

Mr. Connolly said on Thursday that he would recommend that House leaders promptly back the new, higher figures presented by the Postal Service, and Representative Carolyn B. Maloney, Democrat of New York and the chairwoman of the oversight panel, indicated that she saw doing so as a matter of life and death for the agency.

“Unless Congress and the White House provide meaningful relief in the next stimulus bill, the Postal Service could cease to exist,” she said.