Washington (CNN) Senate Majority Leader Mitch McConnell urged the White House to release the rough transcript of the phone call between President Donald Trump and the Ukrainian president to back up Trump's claim that there was nothing improper with the call, according to a source familiar with the conversation.

The White House on Wednesday released the rough transcript of the July 25 phone call between Trump and Ukrainian President Volodymyr Zelensky, a day after House Speaker Nancy Pelosi announced a formal impeachment inquiry into Trump related to the matter. The call summary showed that Trump asked Zelensky to investigate a potential 2020 political rival, Joe Biden, and his son, Hunter.

There is no evidence of wrongdoing by either Joe or Hunter Biden.

The rough transcript led Democrats to accuse Trump of a quid pro quo, after media reports on Monday revealed Trump froze nearly $400 million in military aid to Ukraine days ahead of his call with Zelensky.

Top Democrats concluded the call summary showed clear evidence of Trump pressuring a foreign leader for political advantage.

Trump denied any wrongdoing and argued that the call summary showed there was no direct quid pro quo — an argument that was echoed by his Republican allies in Congress.

"You take a look at that call; it was perfect. I didn't do it. There was no quid pro quo," Trump said in a press conference Wednesday in New York after the rough transcript's release.

On Wednesday, McConnell had dismissed criticism as "laughable to think this is anywhere close to an impeachable offense," Politico reported.

"I've read the summary of the call. If this is the 'launching point' for House Democrats' impeachment process, they've already overplayed their hand," McConnell said in a statement provided to Politico. "It's clear there is no quid pro quo that the Democrats were desperately praying for."

Whistleblower complaint

The July phone call was part of a whistleblower complaint that alleged Trump solicited interference from Ukraine in the upcoming 2020 election.

After reports of the whistleblower complaint emerged, Trump acknowledged that Biden was mentioned on the call with Zelensky and later revealed that he had withheld military aid from Ukraine.

He claimed the delay in aid was not related to the investigation request, but because he was waiting on European allies to pay their fair share.

McConnell on Tuesday said he was " never given an explanation " why the aid to Ukraine was held up even as he was pressing administration officials to release the aid at the time.

Before the July call summary's release, the President had suggested last Sunday that he was open to releasing the call transcript, telling reporters: "I hope they can put it out."

Some of Trump's top administration officials argued otherwise, including Secretary of State Mike Pompeo who said "there's no evidence that would be appropriate here at this point."

"We don't release transcripts very often. It's the rare case," Pompeo said Sunday on ABC News . "Those are private conversations between world leaders, and it wouldn't be appropriate to do so except in the most extreme circumstances."

But two days later, Trump pledged to release a "complete, fully declassified and unredacted transcript of my phone conversation with President Zelensky of Ukraine."

The whistleblower complaint, which was made public Thursday after demands from Democrats, alleged that White House officials tried to "lock down" all records of Trump's Ukraine phone call -- especially the word-for-word transcript produced by the White House.

White House lawyers directed officials to remove the transcript of the call from a computer system that stores them for Cabinet-level officials and instead put the transcript in a system for especially sensitive information, the whistleblower alleged.

The whistleblower said they heard from other White House officials that this was "not the first time" that the Trump administration used this storage system to hold politically sensitive documents. The codeword-level system is meant to hold files of national security importance.

The White House acknowledged Friday that the move to place the Ukraine call transcript in the system came at the direction of National Security Council attorneys.