An email message released Friday shows the Trump administration bracing for scrutiny over the cost of its plan to build a new downtown headquarters for the Federal Bureau of Investigation instead of moving the agency to Maryland or Virginia.

The email raises questions about the White House’s explanation for President Trump’s desire to keep the F.B.I. in Washington. Last month, Sarah Huckabee Sanders, the White House press secretary, said that Mr. Trump “wanted to save the government money.”

The Trump administration has faced questions from congressional Democrats after it abandoned a long-discussed plan to build a campus for the F.B.I. in the Washington suburbs while turning over the bureau’s existing headquarters on Pennsylvania Avenue for commercial development. In February, the administration presented lawmakers with a new plan: It would seek to demolish the existing headquarters, the J. Edgar Hoover Building, and rebuild on that site.

Mr. Trump has taken an unusual interest in the F.B.I. building, including taking part himself in at least one meeting about the headquarters project before the new plan was announced. The Hoover building is a block from the Trump International Hotel, and the plan to build a new F.B.I. headquarters on the existing site will keep the property out of the hands of a commercial developer.

The newly revealed email is from Feb. 13, the day after the Trump administration presented a Senate committee with its plan to keep the F.B.I. on Pennsylvania Avenue. Officials from the General Services Administration, which handles real estate for the federal government, testified later that week in front of a House subcommittee.

In the email, Andrew Abrams, an official at the White House Office of Management and Budget, provided what he described as “the hardest-hitting FBI HQ question we could come up with.”

The email said that the new plan “proposes a less secure facility” and “has a higher per seat cost.” It concluded with the question, “How is this a good deal for FBI or taxpayers?”

The email was released Friday by five House Democrats, including Representative Elijah E. Cummings of Maryland, the ranking member of the House Oversight Committee. Last month, they released emails that shed light on Mr. Trump’s involvement in the F.B.I. project; in response, Ms. Sanders said that the president had wanted to save money.

On Friday, the lawmakers sent a letter to the White House chief of staff, John F. Kelly, seeking documents regarding the F.B.I. project and taking issue with Ms. Sanders’s statement. In addition to Mr. Cummings, the letter was signed by Representatives Gerald E. Connolly of Virginia, Peter A. DeFazio of Oregon, Mike Quigley of Illinois and Dina Titus of Nevada.

“The White House should not be issuing false claims to justify or conceal President Trump’s conflicts of interest on this matter,” the lawmakers wrote.

The White House was reviewing the letter from the lawmakers, a spokeswoman said on Friday.

The cost of the plan to build a new F.B.I. headquarters in downtown Washington, rather than in the suburbs, had previously been in dispute. The new downtown headquarters would house about 8,300 employees, while the planned suburban campus would have housed about 10,600 employees.

When the Trump administration presented its plan to the Senate this year, it listed a price tag of $3.3 billion, compared with $3.6 billion for the plan to build a suburban campus. A report released in August by the inspector general for the General Services Administration challenged that comparison, saying that the new plan would actually be more expensive.