U.S. Attorney John Durham is investigating Ukraine’s role in potential 2016 election interference, the Justice Department confirmed Wednesday, the same day transcripts showed that President Trump’s famous call to the Ukrainian president centered around that country’s possible aid in a probe of a missing Democratic email server.

Attorney General William P. Barr this year tapped Mr. Durham, the top federal prosecutor in Connecticut, to look into the origins of the FBI probe into alleged links between the Trump campaign and Russia during the 2016 presidential race, an investigation that ultimately gave way to special counsel Robert Mueller’s probe.

The Mueller investigation ended earlier this year after concluding that Mr. Trump and key campaign figures did not conspire with Russia to sway the election.

Mr. Durham has been silent about what he has uncovered, but the Justice Department said he has set his sights on Ukraine.

“A Department of Justice team led by U.S. Attorney John Durham is separately exploring the extent to which a number of countries, including Ukraine, played a role in the counterintelligence investigation directed at the Trump campaign during the 2016 election,” Justice Department spokeswoman Kerri Kupec said in a statement.

Ms. Kupec said Ukrainians who are not government officials have volunteered information as part of the probe. She also said Mr. Barr has not contacted the Ukrainian government about the investigation.

The announcement comes as the Justice Department released a transcript of Mr. Trump’s phone call with Ukrainian President Volodymyr Zelensky. In the call, Mr. Trump suggests Ukraine should investigate the business dealings of Hunter Biden, the son of former Vice President Joseph R. Biden, the 2020 Democratic presidential front-runner.

Hunter Biden was on the board of a natural gas company owned by Ukrainian oligarch Mykola Zlochevsky, who was being investigated for corruption. Mr. Zlochevsky was eventually cleared, but some have suggested the probe was not on the level, and that Mr. Biden may have been involved.

It is not clear whether Mr. Durham’s interest in Ukraine is tied to a missing Democratic National Committee server that contains some of Hillary Clinton’s emails.

But that subject, and the role of the cybersecurity firm CrowdStrike, was the start of Mr. Trump’s conversations with his Ukrainian counterpart in the July 25 call, as the now-notorious “favor” he appears to ask Mr. Zelensky centers on whether someone in that country has the server.

“I would like to ask you to do us a favor, though, because our country has been through a lot and Ukraine knows a lot about it. I would like you to find out what happened with this whole situation with Ukraine. They say CrowdStrike. … I guess you have one of your wealthy people … the server, they say Ukraine has it,” Mr. Trump said in the call, according to the transcript.

“I would like to have the attorney general call you or your people and I would like you to get to the bottom of it,” he said.

At that point in the conversation, neither Biden had been mentioned or alluded to.

CrowdStrike is the cybersecurity firm that was hired to probe the hack of the Democratic National Committee’s server in 2016, which yielded a trove of embarrassing emails that were released in batches throughout fall 2016 by WikiLeaks. CrowdStrike traced the DNC server hacks to a group of Russian hackers, and its findings led to Mr. Mueller’s bringing charges against two hacking groups.

The company turned over its forensic analysis of the hacks to the FBI, but it did not provide the server, which has raised eyebrows among conservatives who see the Russia probe as a “deep state” plot against a democratically chosen leader whom it sees as illegitimate.

Mr. Trump has long maintained this theory and tied it to foreign interference, noting among other things that the Democrat-funded dossier by former British spy Christopher Steele was sourced, especially in its most salacious details, to Russians.

Mr. Barr appointed Mr. Durham to investigate the origin of the Russia probe.

It is not known why Mr. Trump appears to believe the missing DNC server is in Ukraine.

Mr. Trump has previously referred to Sunnyvale, California-based CrowdStrike as a Ukrainian company, making the claim in a 2017 interview with The Associated Press.

“I heard it’s owned by a very rich Ukrainian. That’s what I heard,” he said.

CrowdStrike’s co-founder and chief technology officer, Dmitri Alperovitch, is a U.S. citizen and was born in Russia, emigrating after the end of the Cold War.

Some have suggested CrowdStrike helped sweep the DNC breach under the rug. Roger Stone’s attorneys have asked a federal judge to turn over CrowdStrike’s report on the hacks, saying the Justice Department failed to “independently verify” the findings.

The Justice Department has pushed back on those claims. In June, Jessie K. Liu, the U.S. attorney for the District of Columbia who is prosecuting Mr. Stone, said his claims were “mistaken,” “incorrect” and “irrelevant.”

A CrowdStrike spokeswoman said in a statement that the company “provided all forensic evidence and analysis to the FBI.”

“As we’ve stated before, we stand by our findings and conclusions that have been fully supported by the U.S. intelligence community,” said Iliana Cashiola, public relations director for CrowdStrike.

In the conversation with Mr. Zelensky, according to the transcript, the president later offered the help of Mr. Barr again and of his personal attorney Rudolph W. Giuliani to get to the bottom of its Biden probe.

“The other thing. There’s a lot of talk about Biden’s son, that Biden stopped the prosecution and a lot of people want to find out about that. … Biden went around bragging that he stopped the prosecution so if you could look into it. It sounds horrible to me,” he said in the only sentences in the conversation to mention the Bidens, though he later talks about the unfair treatment of the fired prosecutor.

Mr. Biden bragged on camera at a Washington think tank last year about stopping a Ukrainian corruption probe, using a threat to cut off U.S. aid.

“I looked at them and said, ‘I’m leaving in six hours. If the prosecutor is not fired, you’re not getting the money.’ Well, son of a bitch, he got fired,” Mr. Biden recalled telling then-Ukrainian President Petro Poroshenko about prosecutor Viktor Shokin, adding that President Obama was aware of the threat.

Hunter Biden pocketed at least $3 million over five years for a job on the board of Ukrainian gas company Burisma Holdings, despite having no experience in the energy industry. Mr. Shokin has told reporters that Hunter Biden was on a list of Burisma board members he had been preparing to question.

Mr. Trump said in that section of his conversation with Mr. Zelensky that “whatever you can do with the Attorney General would be great.”

But Mr. Barr said that never happened, and he learned of the call only weeks later.

“The president has not spoken with the attorney general about having Ukraine investigate anything relating to former Vice President Biden or his son,” Ms. Kupec said. “The president has not asked the attorney general to contact Ukraine on this or any other matter. The attorney general has not communicated with Ukraine on this or any other subject.”

Despite Mr. Barr’s emphatic denials, at least one powerful Democrat — Rep. Jerrold Nadler of New York, chairman of the House Judiciary Committee — is demanding his recusal from the matter.

“The president dragged the attorney general into this mess. At a minimum AG Barr must recuse himself until we get to the bottom of this matter,” Mr. Nadler tweeted Wednesday.

It is not clear what exactly Mr. Barr would recuse himself from.

The Justice Department criminal review into Mr. Trump’s phone call with Mr. Zelensky was completed and cleared the president. A separate review by the Justice Department’s office of legal counsel completed its legal opinion, concluding the matter was not “urgent” enough to be shared with Congress.

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