"Absent a definitive judgment from the Judicial Branch ... Plaintiff will effectively be forced to adjudicate the Constitutional dispute himself, and if he judges wrongly, he will inflict grave Constitutional injury on either the House or the President," Cooper wrote in a court filing.

Kupperman pointed to what he described as the merits and drawbacks of both the White House and Congress' arguments.

He noted that a court ruled in 2008 that there are limits on claims of "absolute immunity" of presidential advisers to congressional testimony, even if those limits hadn't been tested. That court determined that President George W. Bush's counsel, Harriet Miers and other senior administration officials, did not enjoy "absolute immunity." But the court left the guidelines ambiguous.

"The district court in Miers further concluded that the Counsel to the President was not entitled to absolute or qualified immunity because the inquiry did not “involve the sensitive topics of national security or foreign affairs," Cooper noted. "National security and foreign affairs are at the heart of the information that the House Defendants seek from Plaintiff in connection with the House’s impeachment inquiry."

Former National Security Adviser John Bolton. | Win McNamee/Getty Images

But Kupperman also raised questions about whether the House subpoena itself was valid, in part because of concerns raised by Republicans that the impeachment inquiry itself failed to comply with House rules.

"It is unclear whether a House committee has the authority to issue subpoenas to investigate potentially illegal conduct by an impeachable officer outside the scope of a properly authorized impeachment inquiry," Cooper argues.

In a rebuke of that position, however, a federal judge in a separate matter ruled earlier in the day that the House's impeachment inquiry is valid and constitutional, rejecting the Trump administration's claim that it's an illegitimate exercise of congressional power.

Republicans have contended that the House must hold a formal vote to launch an impeachment inquiry, but in her ruling, Judge Beryl Howell — chief judge of the federal District Court in Washington D.C. — said there is no requirement of a formal vote. Rather, Howell ruled, Speaker Nancy Pelosi's determination that several House committees were pursuing potential impeachment justifies the probe.

It's unclear how Howell's ruling might affect the immediate handling of Kupperman's case.

Democrats, though, cited Howell’s ruling in a Saturday letter to Kupperman blasting his decision to turn to the courts, which they claimed was done in coordination with the White House.

“Dr. Kupperman’s lawsuit—lacking in legal merit and apparently coordinated with the White House—is an obvious and desperate tactic by the President to delay and obstruct the lawful constitutional functions of Congress and conceal evidence about his conduct from the impeachment inquiry,” wrote Reps. Adam Schiff, Eliot Engel and Carolyn Maloney, the three chairs leading the impeachment inquiry.

“Notwithstanding this attempted obstruction, the duly authorized subpoena remains in full force and Dr. Kupperman remains legally obligated to appear for the deposition on Monday.”

“The deposition will begin on time and, should your client defy the subpoena, his absence will constitute evidence that may be used against him in a contempt proceeding,” they added.