A federal judge in Washington said on Monday that the way the Trump administration set up and has been running Elon Musk’s so-called Department of Government Efficiency may violate the Constitution.

The skepticism expressed by the judge, Colleen Kollar-Kotelly, did not come as part of a binding ruling, but it suggested that there could be problems looming for Mr. Musk’s organization, which is also known as the U.S. DOGE Service.

“Based on the limited record I have before me, I have some concerns about the constitutionality of U.S.D.S.’s structure and operations,” Judge Kollar-Kotelly said at a hearing in Federal District Court in Washington. She expressed particular concern that it violated the appointments clause of the Constitution, which requires leaders of federal agencies to be nominated by the president and confirmed by the Senate. Mr. Musk was neither nominated nor confirmed.

Judge Kollar-Kotelly’s remarks about the Musk operation were part of a civil case brought by two labor unions and a group representing millions of American retirees. They are seeking an injunction that would bar the Musk team from accessing sensitive records maintained by the Treasury Department.

Last week, a federal judge in Manhattan, entertaining a similar legal issue, banned Mr. Musk’s cost-cutting group from regaining access to the Treasury Department’s payment and data systems until the conclusion of a separate lawsuit claiming that its access to the records is unlawful.

The suits are among several challenging Mr. Musk’s wide-ranging efforts to scrutinize government spending and slash the federal work force, which have spawned dueling directives from Mr. Musk and the heads of various federal agencies, as well as termination notices that were quickly rescinded.