Mr. Kupperman’s lawyer, Charles Cooper, filed an unusual lawsuit against both the House and Mr. Trump, saying his client was caught between conflicting demands from two branches of government and wanted a judge to tell him which side was right so that he did not inadvertently violate his oath to uphold the Constitution.

“Kupperman is in a classic Catch-22,” Mr. Cooper said, adding, “He cannot satisfy both of them, but he is perfectly prepared to satisfy whichever command this court says is valid.”

Mr. Cooper also represents John R. Bolton, Mr. Trump’s former national security adviser, whom the House is also moving to subpoena, and the judge raised the possibility that Mr. Bolton might file an identical case that would be joined to the Kupperman matter.

Mr. Cooper did not confirm that Mr. Bolton would but said if he sued, the case would raise identical legal issues and would not require more time for briefings and arguments.

Judge Leon also noted that earlier on Thursday, the House voted, largely along party lines, for a resolution setting rules for the coming public phase of the impeachment inquiry and providing the first endorsement by the full chamber for the effort, which the resolution styled as already “ongoing.”

When previously instructing officials to ignore impeachment-related subpoenas, the Trump White House has argued that the lack of such a vote meant the inquiry was a sham. Judge Leon asked a House lawyer, Todd B. Tatelman, whether the committee would issue a new subpoena to Mr. Kupperman in light of the vote. Mr. Tatelman said he was not sure but maintained that the existing subpoenas were lawful.

Mr. Tatelman and Elizabeth Shapiro, a Justice Department lawyer, both told Judge Leon that they would file motions to dismiss the case, arguing that for technical reasons, Mr. Kupperman had no right to sue. Mr. Cooper said he would argue that the lawsuit is valid, but then step back and make no arguments about which branch was right on the legal merits.