Updated Sept. 30, 2019 4:19 pm ET

Secretary of State Mike Pompeo was among administration officials who listened in on the July 25 phone call between President Trump and Ukrainian President Volodymyr Zelensky, a senior State Department official said Monday, a disclosure that ties the State Department more closely to the House impeachment inquiry.

Mr. Pompeo’s participation on the call hasn’t been previously reported. Last week, a State Department official disputed the contention in a complaint filed under federal whistleblower laws by a Central Intelligence Agency officer that another top State Department official, counselor Ulrich Brechbuhl, listened in on the call.

House Democrats have opened an impeachment inquiry focused on Mr. Trump’s dealings with Ukraine, and lawmakers are focusing on the whistleblower complaint and a record of the call released by the administration.

Three House committees on Friday subpoenaed Mr. Pompeo for documents related to the inquiry; the secretary has until Oct. 4 to produce them. Separately, the committees plan to depose five current or former State Department officials regarding Ukraine.

Earlier Monday, Senate Majority Leader Mitch McConnell said he would hold a trial in the Senate if the House approved articles of impeachment for President Trump, addressing doubts he may circumvent Senate procedures.

“I would have no choice but to take it up,” the Kentucky Republican said Monday on CNBC. He added: “How long you are on it is a different matter, but I would have no choice but to take it up based on a Senate rule on impeachment.”

It was the first time Mr. McConnell has directly addressed the topic since House Speaker Nancy Pelosi (D., Calif.) said the House would begin an impeachment inquiry focused on allegations Mr. Trump sought to use the powers of his office to push Ukraine to investigate Democratic rival Joe Biden, and that White House officials acted to conceal evidence of the president’s actions.

The allegations, based on a complaint filed by the CIA officer under federal whistleblower laws and backed by the record of the July 25 phone call between Mr. Trump and Mr. Zelensky, have reverberated internationally.

Mr. Zelensky said Monday that his administration wouldn’t release a transcript of the July phone call with Mr. Trump, while also saying he is open to investigating any alleged violation of Ukrainian law.

Also Monday, a spokesman for Russian President Vladimir Putin reiterated the Kremlin’s position that no phone-call records should be released without mutual consent. The comments came as Democrats demand to know how the White House used a national-security computer server, where records of Mr. Trump’s calls with leaders from Russia and Saudi Arabia—as well as Ukraine—are said to have been stored.

At the White House on Monday afternoon, Mr. Trump told reporters: “We’re trying to find out about a whistleblower.” He did not expand on that, and the White House didn’t immediately respond to a question about the comment.

Mr. Trump has said he deserved to confront the whistleblower and anyone who provided him information and has suggested they are spies who committed treasonous acts. House Democrats are eager to hear testimony from the whistleblower—though in a way that will protect his identity.

Shortly after Mr. Trump’s comment, Andrew P. Bakaj, a lawyer who represents the whistleblower wrote on Twitter: “The Intel Community Whistleblower is entitled to anonymity. Law and policy support this and the individual is not to be retaliated against. Doing so is a violation of federal law.”

The Republican-led Senate is considered unlikely to convict Mr. Trump in any impeachment trial. Removing the president requires approval by two-thirds of the 100-member Senate. Some Senate Republicans have voiced concern over the allegations outlined in a whistleblower complaint made public last week, but none has voiced support for impeachment.

Some GOP lawmakers may want the chance to exonerate the president should the House approve articles of impeachment.

Senate procedures bar the leader from simply ignoring an impeachment vote in the House, based on guidance contained in a 1986 memo from then-Senate parliamentarian Robert Dove.

In the case of President Clinton, the House passed articles of impeachment, and he was acquitted in the Senate. President Nixon resigned before a full vote in the House on articles of impeachment.

Mr. McConnell could move to dismiss articles of impeachment under Senate precedents, as then-Senate Majority Leader Robert Byrd (D., W.Va.) attempted to when Mr. Clinton, who had been impeached by the House, went on trial in the Senate. The dismissal measure failed on a vote of 56-44.

Mr. McConnell also could refer the question of impeachment to a special committee. That is how the impeachment of former President Andrew Johnson was handled, but that occurred before the Senate had rules for handling impeachment. As a result, the parliamentarian wrote, the Senate would need to suspend its rules for impeachment to avoid going to the question of trying someone who had been impeached.

The revelations of the contents of the July phone call between Messrs. Trump and Zelensky thrust the Ukrainian leader into the middle of a partisan fight in the U.S., one of Ukraine’s most important backers financially and politically.

The call record showed that Mr. Trump asked his Ukrainian counterpart to “look into” political rival Joe Biden and his son and said he would direct his personal lawyer Rudy Giuliani and Attorney General William Barr to contact Mr. Zelensky to help him in a possible investigation.

Mr. Zelensky, explaining his reasons for keeping Ukrainian records of the call private, said: “There are certain nuances and things that it would seem incorrect to publish.” Mr. Zelensky spoke to reporters on the sidelines of a military event.

Asked whether Ukraine would open an investigation into the Bidens, Mr. Zelensky said Ukraine would be open to investigate any violations of law. A Ukrainian official earlier this year said he had no evidence of wrongdoing by Mr. Biden or his son Hunter.

“We are open, we are ready to investigate, but this has nothing to do with me,” Mr. Zelensky said. “Our independent law enforcement agencies are ready to investigate any case that violates the law.”

He added: “We can’t be ordered to do anything. We are an independent country.”

In Moscow, Kremlin spokesman Dmitry Peskov said Monday that the Kremlin didn’t believe contents of conversations between Messers. Putin and Trump should be made public without the consent of both parties.

Mr. Peskov told reporters that if “there are signals from the U.S., we will discuss it,” but he underscored that “in general, diplomatic practice, of course, does not provide for such publications.”

The Kremlin and Russia’s Foreign Ministry haven’t been shy about voicing their displeasure over diplomatic exchanges, such as the one between Mr. Trump and Mr. Zelensky, being made public, and have sought to pre-empt any possible request that Mr. Putin’s discussions with Mr. Trump be exposed.

Mr. Peskov said that inquiry under way in the U.S. is “an internal matter.”

“We did not intervene and do not (have the right) to intervene,” he said. “As for the transcript of conversations, of course, to some extent publication is possible, only by mutual agreement of the parties.”

On Sunday, a top aide to Mr. Zelensky, Andrey Yermak, who spoke to Mr. Giuliani soon after the July call, said that Ukraine doesn’t want to get involved in domestic U.S. politics.

“What happens there is their internal political kitchen. We will not take part in this in any way,” Mr. Yermak said in an interview with Ukrainian television.

Mr. Yermak said the Ukrainian administration was preparing for another Zelensky-Trump meeting—this time in an official visit to Washington—without providing more details.

—Alex Leary and Natalie Andrews in Washington and Georgi Kantchev in Kyiv contributed to this article.