The letter was signed by dozens of conservatives, including Ginni Thomas, who is married to the Supreme Court justice Clarence Thomas, Tony Perkins of the Family Research Council and David McIntosh, the president of the anti-tax Club for Growth.

“Conservatives are proud to support the role Mick Mulvaney continues to play in the Trump administration,” they said in the letter, which was initiated by the Conservative Action Project.

The letter continued, “Recent news reports demonstrate that the D.C. swamp is attacking him — and we believe it is because he has been the most successful chief of staff in this administration to advance the Trump pro-America agenda.”

The letter was sent at a moment of peril for Mr. Mulvaney.