Secretary of State Mike Pompeo bucked House subpoenas in the whistleblower probe Tuesday as he accused Democratic committee chairmen of “bullying” his employees, throwing the congressional impeachment inquiry into confusion.

It was the Trump administration’s most forceful counteroffensive yet against the Democrats’ impeachment push and left doubt about whether State Department officials would show up for depositions scheduled for Wednesday at the Capitol.

“I have been made aware that committee staff has been sending intimidating communications to career department professionals,” Mr. Pompeo wrote in a letter to the chairmen. “Let me be clear: I will not tolerate such tactics and I will use all means at my disposal to prevent and expose any attempts to intimidate the dedicated professionals whom I am proud to lead and serve alongside at the Department of State.”

Rep. Adam B. Schiff, California Democrat and chairman of the House Permanent Select Committee on Intelligence; Rep. Eliot L. Engel, New York Democrat and chairman of the House Foreign Affairs Committee; and Rep. Elijah E. Cummings, Maryland Democrat and chairman of the House Oversight and Reform Committee, said Congress would “infer from this obstruction that any withheld documents and testimony would reveal information that corroborates the whistleblower complaint.”

At the White House, President Trump turned up the heat on the anonymous whistleblower by saying he is entitled to confront the individual and the person who gave the whistleblower what the president characterized as “false information.”

“This is simply about a phone conversation that could not have been nicer, warmer or better,” the president tweeted. “No pressure at all (as confirmed by Ukrainian Pres.). It is just another Democrat Hoax!”

The whistleblower, an intelligence official who was working at the White House, lodged a complaint that Mr. Trump pressed the Ukrainian president to dig up dirt on 2020 Democratic presidential candidate Joseph R. Biden and his son Hunter.

The whistleblower complaint is now the crux of Democrats’ reinvigorated impeachment effort.

Sen. Charles E. Grassley, an Iowa Republican who authored many of the nation’s whistleblower laws, broke with the president by saying the person who filed the complaint should be protected.

“We should always work to respect whistleblowers’ requests for confidentiality,” Mr. Grassley said in a statement. “Any further media reports on the whistleblower’s identity don’t serve the public interest — even if the conflict sells more papers or attracts clicks.”

He also cautioned everyone against a rush to judgment.

“Uninformed speculation wielded by politicians or media commentators as a partisan weapon is counterproductive and doesn’t serve the country,” he said.

Mr. Trump has insisted the corruption is on the part of Mr. Biden and his son.

Hunter Biden received $3 million for a job on the board of a Ukrainian energy company while his father, as vice president, pressured Ukraine in 2016 to oust a prosecutor who had been investigating the company.

Mr. Biden said the prosecutor wasn’t pursuing corruption cases, and there is no evidence that Hunter Biden was involved in any wrongdoing.

In response to Mr. Pompeo’s letter, the three chairmen said he was the one intimidating witnesses.

“Any effort to intimidate witnesses or prevent them from talking with Congress — including State Department employees — is illegal and will constitute evidence of obstruction of the impeachment inquiry,” the Democrats wrote.

The vicious exchange raised questions about whether the State Department officials will appear before Congress.

The committees subpoenaed four current State Department employees and one former diplomat.

Mr. Pompeo said the subpoenas, issued late Friday, left the department with no time to prepare. He did not explicitly say whether he would block officials’ testimony.

Former Ukrainian Ambassador Marie Yovanovitch was scheduled to testify Wednesday morning, but her deposition has been postponed until Oct. 11, according to an Intelligence Committee source.

Kurt Volker, the former envoy to Ukraine, is scheduled to testify Thursday as planned. Mr. Volker resigned from his post on Friday after he was named in the whistleblower complaint.

The State Department inspector general on Wednesday is scheduled to brief staffers from several House and Senate committees about documents related to the department and Ukraine.

The meeting was requested by Steve Linick, the State Department inspector general, but scant details are known.

Mr. Trump renewed his attacks on Mr. Schiff. He suggested his critic should be arrested for “fraudulently making up a statement … which was very dishonest and bad for me.”

During a congressional hearing last week, Mr. Schiff was criticized for saying the president asked the Ukrainian government to “manufacture dirt” on the former vice president. Mr. Schiff repeated the claim in a television appearance over the weekend.

But the explosive claim was not part of the whistleblower complaint released last week or the official transcript of the July phone call.

While some in Mr. Trump’s orbit lashed out, another key figure lawyered up. Rudolph W. Giuliani, the president’s personal attorney, hired former Watergate prosecutor Jon Sale to represent him in congressional matters.

House Democrats subpoenaed Mr. Giuliani on Monday for a trove of documents and communications related to his communications with individuals in Ukraine. Mr. Giuliani was named in the whistleblower complaint as “a central figure” in the effort to dig up dirt on the Bidens.

Mr. Sale was involved in the legal battle to obtain secret tapes from President Nixon’s White House to determine whether officials engaged in criminal conduct, according to his biography on the website of his current law firm, Nelson Mullins. Based in Miami, Mr. Sale is co-chairman of the firm’s white-collar and government investigations practice group.

It is not immediately clear whether Mr. Giuliani will comply with the subpoena, which requires him to turn over the materials by Oct. 14. He said Monday that the document request raises “significant issues concerning legitimacy.”

In his phone call with Ukrainian President Volodymyr Zelensky, the president encouraged Mr. Zelensky to connect with Mr. Giuliani and Attorney General William P. Barr on the Biden probe.

Mr. Zelesnky said Tuesday that he has had no interactions with Mr. Giuliani.

“I never met Rudy Giuliani — never. And never had any phone calls with him,” the Ukrainian president said at a news conference in Kyiv.

He insisted he will not be pressured by the U.S. or anyone else to probe the Biden family.

“I don’t feel pressure,” he said at the press conference. “I have lots of people who’d like to put pressure on me here and abroad. I’m the president of an independent Ukraine — no one can put pressure on me.”

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