The most recent federal pay period ended on Saturday, and the government said checks “should be issued at the normal time.” But if the stalemate stretches on, future payments could be delayed or reduced. The District of Columbia’s Department of Employment Services said it expected to receive at least 12 times as many unemployment claims as it normally does.

Worries among federal employees have been intensifying since the shutdown began early Saturday, but Mr. Trump asserted this week, without evidence or elaboration, that many “workers have said to me — communicated — stay out until you get the funding for the wall.”

The claim earned mixed reviews from rank-and-file workers.

“There’s no federal employee who’s going to say, ‘Yes, stop paying me, Mr. President, I don’t mind,’” said Rich Guzofski, who also works for the I.R.S. “To be having this conversation over a wall is stupid,” he added.

Steve Reaves, the president of the union for Federal Emergency Management Agency workers, also cast doubt on Mr. Trump’s claim.

“All of Homeland Security, these are the people President Trump claims to be out there and fighting for,” said Mr. Reaves, who is stationed in the Dallas area. “They’re the people who are being impacted. They’ve got mortgages. They’ve got to still pay bills.”

But a Customs and Border Protection employee in Boston, who did not want her name used because she was not authorized to speak publicly, said that while she was frustrated by the shutdown, she felt that the Democrats were partly to blame and that she supported the president’s push for a border wall.