(CNN) The House of Representatives says it no longer needs to subpoena former deputy national security adviser Charles Kupperman for his testimony, another sign that Democrats don't want or feel they need a prolonged battle to gather more details in their impeachment probe.

"The subpoena at issue in this matter has been withdrawn and there is no current intention to reissue it," the House wrote in a court case filed on Wednesday afternoon about Kupperman's testimony.

The House instead suggested Kupperman should abide by what is decided in a separate case that is further along: its lawsuit against Don McGahn to force the former White House counsel to testify.

The House told Kupperman's attorney on Tuesday it still would be interested in speaking with him in the impeachment inquiry, and that he should sit for a deposition if another federal judge knocks down the White House's blocking tactic in McGahn's case. McGahn was subpoenaed in April to discuss the President's attempts to obstruct the Russia investigation.

McGahn's case is "much closer to resolution by the court than Dr. Kupperman's flawed suit," the House wrote to Kupperman's attorneys on Tuesday. "Unless your lawsuit was admittedly only for purposes of delay, and without a subpoena in force, the Committees expect that your client will voluntarily dismiss the complaint ... and be guided by the decision in McGahn."

A House Intelligence Committee official explained why the House withdrew the subpoena for Kupperman, noting the court process "would only result in delay."

"There is no proper basis for a witness to sue the Congress in court to oppose a duly authorized congressional subpoena," the official said. "Nevertheless, given the schedule of our impeachment hearings, a court process that leads to the dismissal of Dr. Kupperman's flawed lawsuit would only result in delay, so we have withdrawn his subpoena."

"If Dr. Kupperman or other witnesses over whom the White House might assert absolute immunity to prevent their testimony are truly interested in honoring their duty to uphold the Constitution instead of hiding behind the White House's improper directions, they have every opportunity to do so, as numerous current and former officials -- both senior and more junior -- have courageously done."

Kupperman had sued prior to his scheduled testimony to get a court's decision on whether he needed to testify. The White House had told him not to appear before Congress, citing immunity for White House officials.

Kupperman is using the same defense lawyer as former national security adviser John Bolton. The federal judge in Kupperman's case has discussed how Bolton could take the same approach in court, which would delay his own testimony. Bolton is scheduled to testify on Thursday, but it is still unclear whether he will show up if subpoenaed. Bolton's team declined to comment on whether Wednesday's developments affected his approach to his scheduled deposition Thursday.

The House is asking federal Judge Richard Leon to dismiss Kupperman's lawsuit. Leon had planned to consider Kupperman's lawsuit throughout this month and December, effectively sidelining Kupperman and potentially Bolton until he made a decision. Leon planned to speak Wednesday evening with the lawyers from the House, Justice Department and for Kupperman over the phone.

Kupperman was on the line during Trump's July call with Ukrainian President Volodymyr Zelensky. Bolton is at the center of several key events related to the impeachment inquiry, including reportedly raising concerns about Trump's dealings with Ukraine.

Newly released transcripts of House depositions highlighted how several witnesses knew Trump was pushing for political help from Ukraine this summer. They also knew that the US government was holding up aid from Ukraine while also seeking a public announcement that Ukraine would launch investigations into Trump's political rivals, including former Vice President Joe Biden and the Democratic National Committee.