Secretary of State Mike Pompeo played coy when asked at the State Department on Tuesday if he would testify in the House impeachment inquiry into President Donald Trump.

"When the time is right, all good things happen," Pompeo said cryptically during a press briefing on a range of domestic and foreign issues including new sanctions against a Cuban company and the death of an American citizen in Afghanistan.

A reporter had asked the top U.S. diplomat about a series of tweets Trump had sent before the briefing, in which the president said he "would love" to have Pompeo and other officials, including departing Energy Secretary Rick Perry, acting chief of staff Mick Mulvaney, "testify about the phony Impeachment Hoax."

Pompeo has become a significant figure in House Democrats' probe of whether Trump abused his power by pressuring Ukraine to announce investigations into his political rivals in exchange for a White House meeting or possibly a military aid package.

U.S. Ambassador to the European Union Gordon Sondland testified before the House Intelligence Committee that Pompeo was "in the loop" about the pressure campaign.

A spokeswoman for Pompeo said at the time that Sondland "never told Secretary Pompeo that he believed the President was linking aid to investigations of political opponents."

Pompeo is also rumored to be looking to leave the Trump administration soon and run for a U.S. Senate seat in Kansas. Senate Majority Leader Mitch McConnell has publicly encouraged Pompeo to run for the seat, though the State Department has denied he has plans to resign.

Pompeo said Tuesday that he doesn't "have much to say with respect to the Ukraine investigation," other than to assure that the Trump administration has complied as appropriate with the probe.

But he also defended Trump's request for an investigation into a debunked conspiracy theory that Ukraine, not Russia, stole Democratic National Committee emails during the 2016 U.S. presidential election.

"Any time there is information that indicates that any country has messed with American elections, we not only have a right but a duty to make sure we chase that down," Pompeo said. "To protect our elections, America should leave no stone unturned."

Virtually the entire U.S. intelligence community, including a lengthy investigation by former special counsel Robert Mueller, concluded that Russia was responsible for that interference in the election.

Trump, however, has continued to float the so-called Crowdstrike theory that a DNC server has been hidden somewhere in Ukraine. He brought up the conspiracy theory in his July 25 call with Ukraine President Volodymyr Zelenskiy, which led a flood of Democrats to call for impeachment proceedings.

Trump also asked Zelenskiy on that call to "look into" unsubstantiated allegations against former Vice President Joe Biden and his son Hunter, who had served on a Ukrainian gas company when his father had pushed Ukraine to fire a corrupt prosecutor there.

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