(CNN) As President Donald Trump urged yet another foreign government to investigate one of his political rivals -- this time in full view of cameras and microphones -- White House aides and allies of the President were bracing for a bitter standoff with Congress.

White House officials have begun attempts to piece together the timeline and circumstances surrounding the President's now infamous call with Ukrainian President Volodymyr Zelensky, one White House official said.

Meanwhile, White House lawyers were drafting a letter to lawmakers insisting that none of their requests would be fulfilled until the House formally votes to open an impeachment inquiry.

The developments reflect the first outlines of an administration plan to confront the growing impeachment crisis, which has proceeded at rapid pace since details of the President's conversation with Ukraine emerged last week.

Facing an impeachment inquiry focused on his request that Ukraine investigate former Vice President Joe Biden over baseless corruption claims, Trump upped the ante on Thursday and said he would like to see China do the same. Democrats quickly seized on the comments as yet more evidence to bolster their case for impeachment and multiple White House officials and Trump allies privately expressed dismay that Trump was making matters worse for himself.

The comments are at least in part a result of the lack of a formal communications strategy from the White House to handle the impeachment inquiry and a President convinced he is his own best messenger, even when it causes his aides to cringe. The President and top aides have also dismissed the need for a war room to support their response efforts.

"He's going for broke," one person who speaks with Trump said, adding that Trump didn't see his actions as illegal and has theorized his opponents have been doing the same thing to him.

Until now, Trump has appeared to be operating without a plan, lashing out on Twitter and in person in increasingly volatile and bombastic ways. Allies expressed concern that he is without a strategy as Democrats scaled up their efforts.

Those worries persist, particularly as Trump makes clear he sees nothing wrong with soliciting political dirt on his rivals from foreign governments.

But the emerging signs of at least a legal response effort show the White House is taking steps to resist attempts by Democratic lawmakers to dig further into the administration's dealings with Ukraine.

The goal in fleshing out the timeline surrounding the Zelensky call, according to the official, is to ensure the White House has a full picture of the events at hand as it fields inquiries from Congress and prepares to build the President's defense to impeachment. The effort has involved lawyers in the White House counsel's office and Situation Room staff who are involved in connecting the President with foreign leaders and compiling a record of those calls.

Among the facts the officials are looking to compile is a list of individuals involved with the call and those involved in the lead-up and aftermath of the call, though the official said the fact-finding mission is not aimed at uncovering the identity of the whistleblower.

The threat of subpoenas

On Friday, Democrats are expected to issue subpoenas compelling the White House to produce key documents as part of the House of Representatives' impeachment inquiry.

In response, White House lawyers have written a letter arguing they are not compelled to provide documents related to the impeachment inquiry before a formal vote on the inquiry is held, according to a person familiar with the matter.

The letter, though not finalized, could be sent as soon as Friday. It is one of several items under consideration at the White House about how to respond to Congressional investigators. White House officials are still weighing the timing of the letter, which is still being reviewed by White House lawyers.

House Speaker Nancy Pelosi has not yet held a full vote in the House approving of the impeachment inquiry. Trump's allies have argued that means the process has not formally begun.

Giuliani's command of the anti-impeachment narrative

White House officials have been largely sidelined as the President and his personal attorney Rudy Giuliani take command of the anti-impeachment narrative, at times driving the news cycle in ways White House allies believe hurt the President's case -- or, at a minimum, make his position harder to defend -- administration officials and sources close to the White House said.

After Trump unabashedly said he would like Ukraine and China to investigate the man he sees as the biggest threat to his 2020 reelection, it was Giuliani who made matters worse.

In an interview with the Wall Street Journal , he implicated the President directly in the controversial recall of the US ambassador to Ukraine and tied that removal to efforts to get Ukraine to investigate Biden. Multiple sources close to Trump lamented Giuliani's frequent media appearances in the wake of the whistleblower complaint, but said they are powerless as Trump refuses to rein in Giuliani.

For his part, Trump's comments amounted to the latest instance in which he has acted publicly in ways that would be scandalous or improper if done behind closed doors and subsequently uncovered. Just as he called on Russia publicly to find his rival Hillary Clinton's deleted emails and just as many of his public statements and tweets were used by special counsel Robert Mueller as evidence of potential obstruction of justice, Trump on Thursday handed Democrats more evidence of his efforts to get foreign countries to hurt his political rivals.

Forgoing a war room

Trump and top officials, including his son-in-law Jared Kushner, have dismissed the need for a war room to bolster the White House's response to Democrats' impeachment inquiry, as President Bill Clinton's White House established when he faced impeachment.

While the White House counsel's office has been focused on determining how to handle congressional requests for information and subpoenas, other senior White House officials have also had discussions about the House impeachment inquiry.

Officials said White House chief of staff Mick Mulvaney and senior adviser Jared Kushner are the officials most focused on developing an impeachment strategy.

While some said Kushner was in charge of the strategy, others said Mulvaney was at the helm.

"He's the President's senior adviser," one administration official said of Kushner. "Of course he's important, but in no way is he running this."

The official said Kushner has privately advised Trump on the impeachment strategy and has also played a role in overseeing Trump's campaign messaging.

Looking to turn the tables by focusing on unfounded accusations against Biden, Trump tweeted out several political ads attacking the former vice president that were also supported by multi-million dollar ad buys. Kushner had signed off on the ads.