"Rudy very much knows what's happening and he is a very capable guy. If you could speak to him that would be great," Trump tells Ukrainian President Volodymyr Zelensky, according to the memo.

On Tuesday night, Giuliani told Fox News he had gotten in touch with Ukrainian officials at the behest of the State Department. It was one of several times Giuliani has claimed State officials had tasked him with the mission.

"I never talked to a Ukrainian official until the State Department called me and asked me to do it," the former New York mayor told Fox News. He mentioned Kurt Volker, the U.S. special envoy for Ukraine, as one of the State officials who asked for his help.

Democrats jumped on Giuliani’s remarks to demand answers, even as some Hill staffers privately said they are wary of Giuliani.

"Rudy Giuliani needs to explain this under oath. The Senate Foreign Relations Committee should call a hearing ASAP," Sen. Jeanne Shaheen (D-N.H.) tweeted while linking to the clip of Giuliani's interview.

That appeal comes after Sen. Bob Menendez (D-N.J.) sent a letter Tuesday to Pompeo demanding that he hand over records and other material about the department's role in facilitating the Giuliani contacts.

There is no evidence that Hunter or Joe Biden did anything illegal, but the possibility that Trump asked a foreign government to investigate a political opponent prompted House Speaker Nancy Pelosi, who had long seen the impeachment as politically risky, to announce an such an inquiry on Tuesday.

The State Department would not clarify on Wednesday what role Pompeo or his aides played in Giuliani's diplomatic and investigatory efforts.

How to read Trump’s phone call with Ukraine’s president: A line-by-line analysis of the conversation that launched an impeachment inquiry.

While it's still unclear what Pompeo did or did not do, the still-unfolding Ukraine scandal could dramatically affect his standing in Foggy Bottom and on Capitol Hill, not to mention his political ambitions.

Pompeo is mulling a campaign for the Senate from Kansas; he’s also said to be thinking about a future White House run. But his strategy of rarely allowing any public daylight between himself and the president — whatever its effectiveness as a bureaucratic survival strategy — is now proving politically perilous.

“If Trump is discredited inside his own party, it hurts Pompeo,” predicted Thomas Wright, a foreign policy scholar at the Brookings Institution.

State Department staffers, meanwhile, are questioning how the secretary could have allowed the Ukraine-related dealings to go on under his nose and whether he was complicit in derailing the career of a respected ambassador along the way.

“It’s impossible to believe that the secretary wasn’t aware of what was happening,” said one State Department official, speaking on condition of anonymity. “If he was kept in the dark, that’s even more troubling.”

Some observers warn, however, that it's far too soon to tar Pompeo or others in Trump’s circle, and that doing so could backfire on Democrats. After all, the lengthy investigations into Russia’s role in the 2016 presidential election did not yield the incontrovertibly damning evidence that Democrats had hoped.

“My sense is that there is a real danger of Trump’s opponents leaning out over their skis on this stuff. There is no surer way to galvanize the Trump base than to run Russiagate again,” said Ilan Berman, senior vice president of the conservative American Foreign Policy Council.

Lawmakers and their aides are trying to establish what role certain U.S. envoys played in connecting Giuliani with Ukrainian officials, and whether Pompeo signed off.

Those diplomats include Volker and the U.S. ambassador to the European Union, Gordon Sondland. There also are questions about the actions of U.S. ambassadors in other European countries where Giuliani may have met with Ukrainian officials.

Of special interest: the role Pompeo and his aides played in recalling Marie Yovanovitch, the career diplomat who was serving as the U.S. ambassador to Ukraine.

Giuliani, Donald Trump Jr., and conservative media figures had attacked Yovanovitch, alleging, among other things, that she had expressed anti-Trump sentiment.

The ambassador, who has spent decades in the Foreign Service, was yanked from Kiev in May, a couple of months before the expected end of her tenure.

The readout released Wednesday of Trump's call with Zelensky, the Ukrainian president, does not mention Pompeo. But in the memo, both leaders trash Yovanovitch.

“The former ambassador from the United States, the woman, was bad news and the people she was dealing with in the Ukraine were bad news so I just wanted to let you know that,” Trump says.

Zelensky replied that he agrees “100 percent.”

“Her attitude towards me was far from the best as she admired the previous president and she was on his side,” Zelensky said. “She would not accept me as a new president well enough.”

Yovanovitch could not be reached for comment; people who know her say she's not talking to the media. But a former State Department official familiar with her situation said Pompeo's involvement in her case might have been to her benefit. For one thing, she's still on the U.S. payroll.

"My understanding is he stood up for her,” the former State official said. “But he recognizes he’s the secretary and Trump is the president. It might have been worse had Pompeo not been involved. Maybe had he not been, she would have been fired.”

Lawmakers also are trying to establish what role Pompeo played in an earlier holdup of hundreds of millions of dollars in military aid to Ukraine. The Trump administration had placed the financial assistance on hold before Trump spoke to Zelensky and asked him to look into the Bidens.

A person with knowledge of the matter told POLITICO that State Department officials told Senate staffers that Trump’s acting chief of staff, Mick Mulvaney, had told State not to go ahead and obligate the funding to Ukraine.

The State officials said they had no policy objection to handing over the money and that they were not given any reason as to why it should be held up, the person said.

A Senate Democratic aide said the funding was held up for more than two months, and there are questions as to what, if anything, Pompeo did to push it through to the Ukrainians.

A former U.S. official in touch with people in the Trump administration said Pompeo was unhappy with the hold placed by the White House Office of Management and Budget on the Ukraine funding.

By early September, according to an administration official, Pompeo instructed the State Department to blow the hold and just send the money; State’s share of the funding was roughly $140 million, while the Pentagon’s was around $250 million.

State notified lawmakers it was going to send its share of the money around Sept. 11, timing that coincided with the White House's decision to back off its hold on Ukraine funding, according to people familiar with the issue.

Aides to Pompeo declined to comment or did not respond to questions this week. Volker did not respond to requests for comment. Sondland, reached in New York, declined a request for an interview.

Pompeo has largely stayed out of the conversation, with the notable exception of a Sunday talk show appearance in which he appeared to indirectly defend Giuliani’s contacts with Ukrainians. In doing so, the secretary raised an unsubstantiated question of whether Biden had somehow interfered in the 2016 presidential election.

"If there was election interference that took place by the vice president, I think the American people deserve to know," Pompeo said on CBS News.

Pompeo has been among the Trump aides who argued against releasing the notes on his call with the Ukrainian leader, according to a senior administration official. The chief U.S. diplomat has expressed concern about setting a precedent that could affect future conversations with foreign leaders.

Pompeo took over the State Department in April 2018. His ascent pleased many U.S. diplomats who viewed his predecessor, Rex Tillerson, as a management disaster who marginalized career experts in the department.

Pompeo has striven to bring confidence, or “swagger” as he calls it, back to State. At first, he was well-received, but a number of recent personnel moves have undercut his standing.

The moves include Pompeo’s apparent inability to fire Kevin Moley, assistant secretary of State found to have acted abusively and inappropriately toward career staffers.

Pompeo aides insist he’s legally unable to terminate Moley, who has Senate confirmation. But the State Department has failed to respond to several emails from POLITICO over the past month as to whether Pompeo ever asked Trump to oust Moley.

The treatment of Yovanovitch is likely to fuel similar discontent. “It’s disappointing, but not surprising, that another career officer is being mistreated for political purposes,” the State Department official said.

Nancy Cook and David Herszenhorn contributed to this report.