The decision to defy congressional subpoenas opens the two men to possible contempt of Congress charges and illustrates Republican Trump's determination to stonewall the Democratic-led impeachment effort, which threatens to consume his presidency.
The refusal to cooperate could be considered as evidence of obstruction of justice, which would substantiate suspicions about Trump's misconduct, according to an official working on the inquiry who communicated on condition of anonymity.
But other US government officials have not been as reluctant to cooperate.
Lawmakers were hearing closed-door testimony on Tuesday from a senior US diplomat, George Kent, who is one of several officials involved in the Ukrainian matter who have complied with congressional subpoenas.
House Democrats are focusing on Trump's request to Ukrainian President Volodymyr Zelenskiy in a July 25 phone call to look into unsubstantiated allegations about Joe Biden, the former vice president and a leading contender to become the Democratic nominee to run against Trump in the November 2020 US presidential election.
Kent, who has spent much of his career fighting corruption in Ukraine and elsewhere, is the second career diplomat to testify as part of the probe after being subpoenaed. The White House and State Department had ordered them not to appear.
It was unclear what Kent, a deputy assistant secretary of state responsible for US policy toward six former Soviet republics including Ukraine, was telling lawmakers.
According to the New York Times, Kent raised concerns with colleagues as far back as March about Giuliani pressuring Ukraine to pursue investigations into Trump's political rivals.
Marie Yovanovitch, the former US ambassador to Ukraine, on Friday accused the Trump administration in testimony of recalling her in May based on false claims.
Fiona Hill, Trump's former Russia adviser, told lawmakers on Monday that she and her then-boss, former national security adviser John Bolton, were alarmed this summer by efforts to force Ukraine to investigate Biden and other rivals and advised her to notify a National Security Council lawyer, according to a source familiar with her testimony.
Hill told lawmakers that Bolton characterized Giuliani as "a hand grenade who is going to blow everybody up," according to two sources familiar with her testimony. A Bolton spokeswoman said he would have no comment on the testimony.
Gordon Sondland, a Trump political donor appointed as the US ambassador to the European Union who was involved in the Ukraine discussions, is due to testify later in the week in response to a congressional subpoena.
Giuliani had faced a Tuesday deadline to produce the requested documents. His lawyer, Jon Sale, said the documents were protected by legal principles that protect attorney-client communications and presidential communications.
"This appears to be an unconstitutional, baseless, and illegitimate 'impeachment inquiry,'" Sale wrote, echoing language the White House used last week.
Pence's lawyer, Matthew E. Morgan, also cited the White House's response, saying the House had not voted to authorize the inquiry. Democratic leaders have said no vote is necessary.
HUNTER BIDEN DENIES WRONGDOING
The investigation is focused on the July call in which Trump pressed Zelenskiy to have Ukraine officials investigate Biden and his businessman son, Hunter Biden, who had been on the board of a Ukrainian energy company.
Hunter Biden, 49, denied doing anything improper https://www.reuters.com/article/us-usa-trump-whistleblower-biden/hunter-biden-defends-overseas-work-in-interview-but-expresses-regret-idUSKBN1WU1PS in past work for the company but acknowledged he had benefited professionally from his father's political career in an interview with ABC News that aired on Tuesday.
Joe Biden has denied Trump's allegations of corruption.
Democrats have accused Trump of pressuring a vulnerable US ally to dig up dirt on Biden after withholding $391 million in US security aid intended to help combat Russian-backed separatists in the eastern part of Ukraine. Zelenskiy agreed to investigate. Trump eventually allowed the aid.
Trump has denied wrongdoing and defended his request to Zelenskiy during the phone call, which prompted a formal whistleblower complaint by a person with a US intelligence agency.
If the Democratic-controlled House votes to approve articles of impeachment - formal charges - the Republican-controlled Senate would then hold a trial on whether to remove the president from office.
According to State Department emails seen by Reuters, diplomat Kent told colleagues that Yovanovitch had become the target of a "classic disinformation operation." Yovanovitch on Friday denied Giuliani's allegations that she provided a "do not prosecute list" to Ukrainian officials to protect Biden and others.
"One key sign of it being fake is that most of the names are misspelled in English - we would never spell most that way," Kent said in the email to colleagues.