In a letter to the House of Representatives on Tuesday -- the deadline set by the subpoena for Pence to hand over the records -- Matthew Morgan, the vice president's counsel, repeated the Trump administration's view that the ongoing impeachment inquiry led by House Democrats is "unfair" and lacks "due process."

Morgan wrote in the letter that "the purported 'impeachment inquiry' has been designed and implemented in a manner that calls into question your commitment to fundamental fairness and due process rights," and Pence's office will cooperate on the investigation only when it is conducted "in a manner consistent with well-established bipartisan constitutional protections and a respect for the separation of powers."

House Speaker Nancy Pelosi announced the initiation of a formal impeachment inquiry into President Donald Trump on Sept. 24 over the president's conduct during a controversial phone call with Ukrainian President Volodymyr Zelensky on July 25.

The impeachment inquiry was prompted by a whistleblower complaint filed in late August alleging that Trump pressured Zelensky to investigate Democratic presidential candidate Joe Biden and his son Hunter Biden, whom Trump accused of involvement in a corruption case in Ukraine.

One of the reasons the White House considers the impeachment proceedings illegitimate is the absence of a full House vote.

"Never before in history has the Speaker of the House attempted to launch an 'impeachment inquiry' against a President without a majority of the House of Representatives voting to authorize a constitutionally acceptable process," read the letter.

Pelosi said at a news conference Tuesday evening that she will not hold a vote on the House floor to authorize an impeachment inquiry as of now. "We're not here to call bluffs," she said. "This is not a game to us. This is deadly serious."

Standing by the side of Pelosi and House Intelligence Committee, Chairman Adam Schiff said "the Constitution is very clear: The House will have the sole power of impeachment."

"Republicans know it, but they don't want to discuss the president's conduct," Schiff added. "They'd rather discuss the process."

Pelosi and three House committees leading the impeachment inquiry -- Intelligence, Oversight and Reform and Foreign Affairs -- believe Trump's conversation with Zelensky, as well as administration officials' interactions with the Ukrainian authorities before and after the call, manifested "efforts to pressure foreign powers to intervene in the 2020 elections."

Trump has denied wrongdoing, insisting that the call was "perfect" and calling the impeachment inquiry "another witch hunt" against him.

Also on Tuesday, Rudy Giuliani, Trump's personal lawyer who has been at the center of the controversy and who has also been subpoenaed for documents, refused to comply with Congress' demand as well.