Ms. Newbold, a 39-year-old employee who has worked in the White House for 18 years under both Republican and Democratic administrations, said she decided to go to the committee after attempts to raise concerns with her superiors and the White House counsel went nowhere, according to the Democratic staff’s account.

“She’s begging us to do something,” Mr. Cummings said during the meeting. “Because she simply wants her government to work the way it’s supposed to work.”

Republicans on the committee argued against the subpoena because Mr. Kline’s lawyer indicated that Mr. Kline was willing to appear before the panel voluntarily. In their own memo summarizing Ms. Newbold’s comments to the committee, they presented her concerns as overblown and depicted her as a disgruntled federal employee.

“We’re going to subpoena a guy who just sent us a letter saying he’s willing to come here voluntarily,” said Representative Jim Jordan of Ohio, the committee’s top Republican. “I’ve been on this committee 10 years; I’ve never seen anything like this.”

But Mr. Cummings said that Mr. Kline, who now works at the Department of Defense, refused to answer questions from the committee about specific federal officials and clearance decisions during a voluntary appearance. Mr. Cummings said that the White House refused two committee requests to interview Mr. Kline and that Mr. Kline did not respond to two separate requests sent directly to him since January.

In a letter to Mr. Cummings, dated Tuesday, Mr. Driscoll wrote that his client was willing to appear but that the White House Counsel’s Office deemed some of the proposed topics as an infringement upon President Trump’s rights as the chief executive. Mr. Driscoll called a subpoena an “extreme and unnecessary step.”

Mr. Cummings said he planned to seek testimony from other senior White House officials as well.

Most of the Republican members of the committee also opposed the majority’s proposal to subpoena testimony from a top Justice Department official, John Gore, about adding a citizenship question to the census.