President Trump’s administration deepened its pressure campaign on government employees to resign before a Thursday deadline, rattling and angering a civil service steeling itself for a prolonged battle with Elon Musk and his ongoing foray into the federal bureaucracy.

With hours dwindling for workers to decide whether or not to quit, agency officials held last-minute meetings to walk their teams through a dizzying barrage of emails detailing the offer to leave their jobs — and to urge them to take the deal.

The resignation push is part of a broader campaign by Mr. Trump and Mr. Musk to drastically cut the size of the federal government. On Wednesday, Mr. Musk proudly proclaimed his team’s advance into another federal agency, announcing that his so-called Department of Government Efficiency, an arm of the White House overseeing downsizing efforts, would “make rapid safety upgrades” to the nation’s air traffic control system.

The announcement came as investigators continued their probe into the Jan. 29 crash of an Army Black Hawk helicopter and an American Airlines passenger jet that killed 67 people.

Mr. Musk’s team was also spotted at the National Oceanic and Atmospheric Administration, where they obtained access to the agency’s computer systems to search for programs and staff tied to diversity policies that the Trump administration has vowed to stamp out, according to people familiar with the situation and screenshots viewed by The New York Times.