Washington (CNN) The House still seeks former White House counsel Don McGahn's testimony and secrets from former special counsel Robert Mueller's investigation, saying that Mueller's work is relevant in a Senate trial of President Donald Trump and that the President could face more articles of impeachment in the future because he is still under investigation

The arguments made to an appeals court on Monday amount to a broad catalogue of Democrats' thinking about the two historic investigations of the President -- the Mueller investigation and the Ukraine investigation that led to the President's impeachmen t -- and how they fit together.

The arguments come as Democratic members of Congress publicly push the White House to turn over more documents and to allow witnesses to appear for testimony during a Senate trial. The Trump administration and Republican Senate leadership have refused both requests.

"The Committee's investigations did not cease with the House's recent impeachment vote," the House wrote to an appeals court on Monday.

Both the House and the Justice Department on Monday wrote a series of arguments to the circuit court, addressing how the impeachment of the President affects both the House's need for McGahn's testimony and the Mueller details.

The Justice Department, representing the White House and McGahn as well as its own interests, has broadly argued that the courts shouldn't step in to give the House the information it seeks.

The McGahn case

The fight over McGahn's testimony could either lead appeals courts to strike down or endorse the White House's claims of absolute immunity for former and current officials who receive congressional subpoenas. That type of ruling could have major implications for a Senate trial, no matter its focus.

The Democratic-led House Judiciary Committee said it has needed McGahn to testify and that it needed to review the Mueller grand jury information as part of possibly impeaching Trump for obstruction of justice.

But Trump hasn't been impeached for obstruction of justice -- such as on the findings from Mueller that he attempted to obstruct the Russia investigation that came after McGahn gave key statements to criminal prosecutors.

Trump wasn't charged with a crime, though Mueller outlined several legal reasons why Trump allegedly obstructed justice. Instead, the attorney general and then-deputy attorney general chose not to prosecute Trump.

The House voted to impeach Trump for abuse of power related to his pressure on Ukraine for a political favor and for obstructing Congress, including not allowing former White House officials to testify, just like the White House has done with McGahn.

McGahn has refused to testify to the House Judiciary Committee since April about what he witnessed Trump say and do regarding the Russia investigation. Mueller reported that McGahn spoke to prosecutors and the FBI several times about Trump's direction to fire Mueller, among other things.

"Understanding President Trump's history of obstruction would be important to the House's presentation of the evidence and the Senate's consideration of the second Article of Impeachment," obstruction of Congress, the House wrote to the DC Circuit Court on Monday.

"For example, if McGahn confirms to the Committee that the President ordered him to fire Special Counsel Mueller—an event that President Trump has publicly disputed— and then tried to cover it up, that testimony would constitute powerful evidence of the pattern of obstructive behavior described in the second Article," the House writes in the new filing in DC Circuit Court.

Separately, "If McGahn's testimony produces new evidence supporting the conclusion that President Trump committed impeachable offenses that are not covered by the Articles approved by the House, the Committee will proceed accordingly—including, if necessary, by considering whether to recommend new articles of impeachment," the House wrote.

If McGahn were ordered by a court to appear for congressional testimony under a subpoena, he may still be able to use the President's assertions of executive privilege to decline to answer some questions.

The Justice Department, representing McGahn, countered on Monday that the courts should stay out of the dispute.

It's especially problematic for the courts to resolve the testimony standoff, the Justice Department argues, while the Senate tries the President for obstructing Congress.

"Indeed, if this court now were to resolve the merits question in this case, it would appear to be weighing in on a contested issue in any impeachment trial. That would be of questionable propriety whether or not such a judicial resolution preceded or post-dated any impeachment trial," the Justice Department wrote just after midnight early Monday. "The now very real possibility of this Court appearing to weigh in on an article of impeachment at a time when political tensions are at their highest levels -- before, during, or after a Senate trial regarding the removal of a President -- puts in stark relief why this sort of interbranch dispute is not one that has 'traditionally thought to be capable of resolution through the judicial process.'"

Politically, more articles of impeachment against Trump seem unlikely at this time.

But in court, the House has argued to keep the McGahn case alive so it can keep its investigative options open and gather more information during the Senate trial.

The Mueller details case

On Monday, the House committee wrote that it wants the information collected by Mueller's grand jury for the upcoming Senate impeachment trial. The information may speak to a "pattern of soliciting foreign election interference and obstructing government investigations of his misconduct."

The House added on Monday that its Judiciary Committee is considering legislation that could change how the White House and Justice Department interact regarding civil and criminal matters. The committee is also considering placing reporting requirements on "foreign offers of assistance," according to the filing.

But the Trump administration said the House should no longer need grand jury secrets following its impeachment vote last Wednesday.

Trump was impeached for his dealings with Ukraine and not his attempts to obstruct the Mueller investigation or other Russia-related findings, the Justice Department argued in another filing to the DC Circuit Court of Appeals Monday.

"Neither article of impeachment adopted by the House, however, alleges high crimes or misdemeanors stemming from the events described in the Mueller Report," the Justice Department writes in its latest filing. "Accordingly, nothing appears to remain of the Committee's alleged need for the grand-jury materials in the Mueller Report."

Previously, the House highlighted that it wants the grand jury details, such as testimony from the convicted former Trump campaign chairman Paul Manafort, because transcripts and other documents could show Trump lied to Mueller about his knowledge of his political adviser's efforts with WikiLeaks during the campaign.

The court filings on Monday come in two of the most potentially significant court cases that could be resolved during Trump's impeachment proceedings. Both the McGahn and grand jury secrets cases are set to be heard by three-judge panels on January 3 at the DC Circuit Court of Appeals in Washington.

The House won both cases in the lower court.

Historically, federal courts try to stay out of clashes between the executive branch and Congress, or Congress and the Justice Department have negotiated before an appeals court settled disputes over administration witnesses. Yet a trial-level judge in Washington, Judge Ketanji Brown Jackson, ruled in late November that McGahn would have to testify. The White House couldn't claim a blanket immunity to override a congressional subpoena of a former official, the judge wrote.

Separately, Chief Judge Beryl Howell of the DC District Court wrote that the House's investigation of the President was valid, and in some ways continued the work of the Mueller grand jury. Howell ruled that the House could view the secret Mueller grand jury information.