WASHINGTON — House Democrats on Monday subpoenaed President Trump’s private lawyer, Rudolph W. Giuliani, a key figure in their impeachment inquiry, even as the president vowed to learn the identity of the anonymous whistle-blower whose accusations of presidential impropriety toward Ukraine lie at the center of the growing scandal.

The subpoena of Mr. Giuliani, who has acted as Mr. Trump’s personal representative in Ukraine, punctuated what was another swift-moving day of confrontation in the capital, rife with accusations by both Democrats in Congress and an increasingly combative president.

“Our inquiry includes an investigation of credible allegations that you acted as an agent of the president in a scheme to advance his personal political interests by abusing the power of the office of the president,” three Democratic House chairmen wrote in a letter to Mr. Giuliani accompanying the subpoena.

Democrats also requested documents and testimony from three of Mr. Giuliani’s associates said to be connected to an effort to pressure the Ukrainians into investigating Democratic rivals of Mr. Trump and promised more subpoenas for other witnesses in the coming days.

Mr. Trump continued his attack on the anonymous whistle-blower whose complaint helped mobilize House Democrats, as well as on Representative Adam B. Schiff, the chairman of the House Intelligence Committee, who is leading the impeachment inquiry.

The president said that the whistle-blower “knew almost nothing” and that the White House was “trying to find out” the whistle-blower’s identity — an action legal experts said could constitute an illegal reprisal. And he went as far as to question whether Mr. Schiff ought to be arrested for treason — a new extreme even for a president who has shredded modern political rules.

“This whole thing has been a disgrace,” Mr. Trump told reporters in the Oval Office.

As quickly as Mr. Trump and his Republican allies leveled criticism at the investigation, other potential leads were emerging related to the July phone call with President Volodymyr Zelensky of Ukraine in which Mr. Trump pushed for a corruption investigation into former Vice President Joseph R. Biden Jr., a leading Democratic presidential challenger next year, and previously unreported attempts by Mr. Trump to enlist investigative help from other foreign governments.

Secretary of State Mike Pompeo was identified for the first time as being among the government officials listening in on the July call between the American and Ukrainian leaders. Just last week, Mr. Pompeo told reporters in New York that “each of the actions that were undertaken by State Department officials was entirely appropriate” in dealing with Mr. Zelensky’s government. Mr. Pompeo’s involvement was first reported by The Wall Street Journal, and could soon prompt another subpoena for testimony.

And if it was not already clear that the impeachment effort against Mr. Trump will not be resolved anytime soon, Senator Mitch McConnell, Republican of Kentucky and the majority leader, said on Monday for the first time since the Ukraine revelations burst into public view that if the House impeached the president, a Senate trial on whether to convict Mr. Trump would be unavoidable.

“I would have no choice but to take it up,” Mr. McConnell said on CNBC.

For now, though, Democrats remain tightly focused on Ukraine and what efforts were made there on behalf of Mr. Trump. In targeting Mr. Giuliani, Democrats are aiming directly at the man who appears to be at the center of the pressure campaign against the Ukrainian government.

Mr. Giuliani is mentioned frequently in the whistle-blower’s complaint and in recent weeks, he has admitted in interviews to trying to gather dirt in Ukraine on Democrats, including Mr. Biden, that would help the president politically. He has also implicated State Department officials in his work.

Mr. Giuliani has now been asked by House investigators for a wide range of communications and documents going back to January 2017, including anything related to $391 million in American security aid that the Trump administration temporarily withheld from Ukraine and that Democrats have speculated may have been part of an effort to pressure the Ukrainian government for information.

Democrats also requested documents and depositions in the coming two weeks from three associates of Mr. Giuliani: Lev Parnas, Igor Fruman and Semyon Kislin.

Mr. Parnas, who has known Mr. Giuliani for years, worked with Mr. Fruman to gather information on the ground in Kiev about the Bidens and Mr. Trump’s former campaign chairman, Paul Manafort, and he helped connect Mr. Giuliani and Ukrainian prosecutors, as The New York Times reported in May.

Mr. Parnas and Mr. Fruman are executives of an energy company that donated $325,000 to a pro-Trump “super PAC” last year, prompting a Federal Election Commission complaint by a nonpartisan campaign finance watchdog accusing the men and the company of violating campaign finance laws.

In an interview, Mr. Giuliani did not say whether he planned to comply with the subpoena. He denounced Democrats, accusing them of issuing an overly broad request. “We’re getting really close to, if we haven’t met, the standard of the McCarthy hearings where nobody seems to care about things like attorney-client privilege,” he said.

Mr. Giuliani has discussed having a longtime associate, Jon A. Sale, an assistant special prosecutor during Watergate, assist him as he weighs how to respond to the subpoena. But in the interview, he said that Mr. Sale was simply a friend, and that he had not chosen a lawyer yet.

Like a growing list of potential witnesses, he will now face an excruciating choice: Either he can hand over material under subpoena that could help build the case against the president, or refuse and help build a possible impeachment article based on obstruction of Congress. The chairmen gave Mr. Giuliani about two weeks, until Oct. 15, to comply.

“Your failure or refusal to comply with the subpoena, including at the direction or behest of the president or the White House, shall constitute evidence of obstruction of the House’s impeachment inquiry and may be used as an adverse inference against you and the president,” they wrote.

Democratic investigators are moving quickly, buoyed by new national polls that suggest a majority of the public, if few Republicans in Congress, supports their work.

In the last few days alone, they have reached a deal to privately meet with the whistle-blower, though no date has been set. They have scheduled depositions beginning this week with State Department officials with knowledge of the accusations. And on Friday they plan to hold a closed hearing with the intelligence community inspector general, who investigated the whistle-blower’s complaint.

Mr. Trump’s attacks on the whistle-blower and Mr. Schiff appear to be intended not only to discredit the growing claims against him but to put a chill on the investigation itself.

They come as White House officials are divided over how best to approach a modern impeachment inquiry. The president is increasingly frustrated with the leadership provided by Mick Mulvaney, his acting chief of staff, and has been relying on other advisers. He relies the most on Mr. Giuliani, according to four people who have spoken with him.

While the whistle-blower has an expectation of confidentiality from the inspector general, some lawyers warn that it is not always absolute. There is nothing in the statute directly prohibiting Mr. Trump from directing his staff to learn the identity of the whistle blower, said Dan Meyer, the former executive director of the intelligence community whistle-blowing program and managing partner at the law firm Tully Rinckey’s Washington office.

But the whistle-blower is protected against reprisals, and any such effort by the White House could become part of demonstrating an intent to carry out a reprisal.

“A threat is reprisal,” Mr. Meyer said on Monday. “The president has threatened execution, so the president has already reprised.”

Mark Zaid, a lawyer for the whistle-blower, took a different view.

“As the acting D.N.I. testified last week, the law and policy supports protection of the identity of the whistle-blower from disclosure and from retaliation,” he said, referring to the acting director of national intelligence. “No exceptions exist for any individual.”

Mr. Trump’s suggestion that Mr. Schiff could be arrested for treason because he paraphrased the president’s July call with Mr. Zelensky to highlight what he regarded as its implicit threats in a way that arguably exaggerated Mr. Trump’s words was equally unprecedented.

“Shorn of its rambling character and in not so many words, this is the essence of what the president communicates,” Mr. Schiff said in prefacing the remarks during a congressional hearing last week.

Mr. Trump has defended his part of the conversation with Mr. Zelensky, the president of Ukraine, as “perfect,” and focused on Mr. Schiff’s public retelling of the call.

Reporting was contributed by Kenneth P. Vogel, Michael S. Schmidt, Eileen Sullivan, Maggie Haberman and Lara Jakes.