President Trump has been candid for months about his intent to have Attorney General William Barr investigate the overseas origins of the FBI’s counterintelligence probe into the 2016 Trump campaign, a move that now is adding to Democrats’ clamor for impeachment.

“I hope he looks at the U.K., and I hope he looks at Australia, and I hope he looks at Ukraine,” Mr. Trump told reporters May 24. “I hope he looks at everything, because there was a hoax that was perpetrated on our country. And somebody has to get to the bottom of it.”

Mr. Barr also has been forthright for months about his desire to probe how and why the FBI during the Obama administration obtained permission from a secret court to monitor Trump campaign officials in 2016.

“I think spying on a political campaign is a big deal,” Mr. Barr told Congress during a public hearing in April. “I am not suggesting that those rules were violated, but I think it’s important to look at them.”

Although Democrats and much of the media are criticizing Mr. Barr’s involvement, many legal specialists say there’s nothing inappropriate about the president dispatching the attorney general to find out what led to special counsel Robert Mueller’s investigation, which found no collusion between Russia and the Trump campaign to interfere in the presidential election.

“I don’t think there’s anything remotely criminal here,” former Justice Department prosecutor Jim Trusty said on Fox News. “I don’t think we’re talking about something that you could turn into a federal offense. There’s no real violation. I wish the president wouldn’t specifically mention [Democratic front-runner Joseph] Biden … but sometimes he pushes that a little bit far.”

In a July 25 phone call at the heart of the impeachment inquiry, Mr. Trump pressed Ukraine President Volodymyr Zelensky for an investigation of Mr. Biden and his son Hunter, who had been paid $3 million by an Ukrainian energy company while his father was vice president.

Mr. Trump also asked Ukraine’s leader to work with Mr. Barr on the origins of the Russia probe, some of which centered on the FBI’s digging into former Trump campaign manager Paul Manafort’s work in the country.

“I think it’s legitimate for them to look at the FISA [foreign intelligence surveillance act] process,” said a veteran Washington defense lawyer with experience in such matters. “Barr announced when he came in that he was going to look into the use of FISA warrants, and whether they were appropriately initiated and conducted. In some sense, this is part of that.”

Current and former Justice Department officials have said there’s nothing illegal or unethical about Mr. Trump asking world leaders to cooperate with the attorney general and that Mr. Barr has the authority to speak with foreign law-enforcement officials about U.S. investigations.

Ron Aledo, a former CIA analyst and Homeland Security employee who has worked in several U.S. embassies, said Mr. Trump’s phone call on July 25 with Ukraine’s leader “is exactly what any given Department of Justice attache does any given day in all U.S. embassies around the world: request the host country for information and updates on ongoing investigations.

“If Trump did something illegal, then all the DOJ attaches worldwide do something illegal every single day,” Mr. Aledo said. “They request criminal intelligence exchanges. If you have any evidence you want to share with me, we have a reason we want to share with you.”

Mr. Trump sought help in the probe of the FBI investigation from Australia, but his phone call to that country’s prime minister actually was preceded by an Australian offer of help on May 28. Its ambassador to the U.S., Joe Hockey, wrote to Mr. Barr to offer Australia’s assistance.

“The Australian government will use its best endeavors to support your efforts in this matter,” Mr. Hockey wrote, referring to the president’s announcement four days earlier to investigate the origins of the FBI investigation.

Still, some say the involvement by Mr. Barr is casting him in a partisan light.

“Trump didn’t designate Barr to play some neutral, objective role,” said Charles Tiefer, a law professor who specializes in the legislative branch at the University of Baltimore. “Trump designated Barr to bring back ‘the goods’ on the Democrats — inculpatory evidence about the Democrats.”

Rep. Don Beyer, Virginia Democrat, said on Twitter that Mr. Barr “is billing US taxpayers to fly all over the world asking foreign countries to intervene in our domestic politics by attacking US intelligence findings in order to massage the President’s fragile ego.”

People for the American Way President Michael Keegan said Mr. Barr’s task “unfortunately looks for all the world like an attempt to discredit our intelligence community’s findings that Russia interfered in the 2016 election.”

“Attorney General Barr continues to approach his office as if he were the president’s lawyer rather than the lawyer for the American people, which the attorney general should be,” Mr. Keegan said. “The latest developments are just one more indicator that his Justice Department has been completely co-opted to serve Donald Trump’s political agenda.”

Senate Judiciary Committee Chairman Lindsey Graham, South Carolina Republican and a Trump ally, disagrees with that view.

“It is clear to me that liberals are starting an attack against Barr to shut him down from going to Italy, Australia and the U.K. to find out if there was something wrong regarding opening up the investigation of the Trump campaign to begin with,” Mr. Graham told Fox News. “If that’s not a double standard, I don’t know what would be.”

He said the foreign governments “should be cooperating with Barr.”

“There’s nothing wrong with Barr looking into this,” he said. “As matter of fact, it would be wrong if he didn’t.”

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