"We're not fooling around here", Schiff told reporters.
Pompeo is under increasing scrutiny from House Democrats leading impeachment proceedings against Trump.
Ukraine's former president said Wednesday he discussed investments with President Donald Trump's attorney, Rudy Giuliani, in 2017, but that he never discussed Ukrainian companies with any us official.
The White House on Wednesday released an unclassified version of the transcript of the phone call.
Pompeo has tried to delay five current and former officials from providing documents and testimony in the inquiry that could lead to charges against Trump.
Trump was highly critical though of the whistleblower report alleging he used the power of his office to solicit interference from a foreign country in the 2020 USA election.
Early Wednesday morning, Pompeo confirmed that he was on the July phone call with Trump and Ukraine's Volodymyr Zelensky after initially playing down any knowledge of what was discussed on the call.
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Volker played a direct role in trying to arrange meetings between Rudy Giuliani, who is Trump's personal lawyer, and Zelensky, the chairmen said.
Trump made his request to Zelenskiy shortly after he had frozen almost $400 million in USA aid to Ukraine.
Among the documents being sought are any communications among administration officials involving Trump's phone call with Zelensky. The aid money was later provided to Ukraine.
He said the "impeachment nonsense" was going nowhere and accused Democrats of deliberately using it to drive the stock market down in the hope that a weaker economy would help them win the 2020 election.
Trump, who is seeking a second four-year term as president, has denied wrongdoing and has assailed the impeachment probe. Will Trump be impeached by the House? He also said he "might" limit how many people listen in on his calls with foreign leaders.
"I'm concerned with aspects of the Committee's request that can be understood only as an attempt to intimidate, bully, & treat improperly the distinguished professionals of the Department of State, including several career FSOs", Pompeo posted on Twitter.
Pelosi, who announced last week the House was conducting an "official" impeachment inquiry into Trump, opened the press conference by talking about legislation Democrats are crafting to address prescription drug prices.
Pompeo said State Department employees had been contacted directly by lawmakers or their staff and told not to talk to the State Department's legal counsel. He has said, however, that he would cooperate with Congress.
The Oversight panel will issue the subpoena on Friday in co-ordination with the House Intelligence and Foreign Affairs committees, Mr Cummings said. Atkinson has concluded that the whistleblower complaint was of urgent concern and appeared credible.