By Dustin Volz, Wall Street Journal–

Attorney General William Barr said Monday he didn’t expect former President Barack Obama or Vice President Joe Biden to be investigated as part of an examination of the origins of a federal probe into whether the 2016 Trump campaign had colluded with Russia.

Mr. Barr last year appointed John Durham, the U.S. attorney for Connecticut, to review the origins of the 2016 probe into possible links between the Trump campaign and Russian election interference. The investigation, according to people familiar with it, is proceeding on multiple fronts, examining how the initial allegations surfaced in 2016 as well as a separate 2017 U.S. intelligence report that concluded Moscow interfered in the presidential election in part to help then-candidate Donald Trump.

“As to President Obama and Vice President Biden, whatever their level of involvement, based on the information I have today, I don’t expect Mr. Durham’s work will lead to a criminal investigation of either man,” Mr. Barr said in response to a reporter’s question during a news conference called to discuss updates to the investigation of a shooting at a military base last year in Pensacola, Fla. “Our concern over potential criminality is focused on others.”

A spokesman for Mr. Obama declined to comment. Mr. Biden didn’t immediately respond to a request for comment.

Mr. Barr didn’t provide details on exactly what or whom Mr. Durham was investigating, but he expressed concern generally about a trend to “gin up allegations of criminality by one’s political opponents based on the flimsiest of legal theories.”

He didn’t directly address remarks made in recent weeks by President Trump and some of his conservative allies that have suggested without evidence that Messrs. Obama and Biden, the presumptive Democratic presidential nominee and Mr. Trump’s expected 2020 opponent, engaged in criminal acts to spy on his 2016 election campaign.

But Mr. Barr said, as he has in the past, that Mr. Trump was the victim of a yearslong “utterly false Russian collusion narrative” and that standards at the Justice Department were abused to reach a particular result. “We can’t allow this to ever happen again,” he said.

Speaking at the White House Monday afternoon, President Trump said he was a “little surprised” by Mr. Barr’s announcement about the former president and vice president, adding that “if it was me, I guarantee they’d be going after me.” He said he had “no doubt” that Messrs. Biden and Obama were involved in what he called the “takedown of a president.”

Still, the president called Mr. Barr an “honorable man” and said he would rely on him to “make all of those decisions.”

Mr. Barr’s comments came amid controversy over the Justice Department’s move to drop its criminal case against Michael Flynn more than two years after the former national security adviser pleaded guilty to lying to the Federal Bureau of Investigation. The White House announced Monday that the interim U.S. attorney for the District of Columbia who signed off on the reversal would be nominated to head the U.S. Drug Enforcement Administration. Justin Herdman, the top federal prosecutor for the Northern District of Ohio, will be named to be the U.S. Attorney in D.C., and the DEA’s current administrator, Uttam Dhillon, will depart for a role at the Justice Department’s main headquarters. It wasn’t clear when the moves would take effect.

The Justice Department’s action in the Flynn case still needs approval from a judge, who last week appointed a retired federal judge to argue against the government’s bid to dismiss and further examine whether the retired three-star general might have perjured himself again during the course of the proceedings.

The Justice Department inspector general released a report in December 2019 finding that the FBI had sufficient reason to open an investigation into the Trump campaign and wasn’t motivated by political bias, but that the bureau didn’t have adequate grounds to keep wiretapping a former Trump aide after a certain point.

Special counsel Robert Mueller’s 2019 report said there were repeated contacts between Russia-linked entities and Trump campaign officials, but investigators didn’t establish that anyone affiliated with the GOP presidential campaign knowingly conspired with Russian interference efforts.

Mr. Durham’s probe would determine whether any federal laws were broken during the Russia investigation, Mr. Barr said. However, citing a Supreme Court decision throwing out the so-called Bridgegate convictions last week, Mr. Barr said there was a difference between an abuse of power and a federal crime and that not all abuses met the threshold of criminality.

Mr. Barr said it was important the American public would be able to vote in November for a presidential candidate “based on a robust debate of policy issues.”

“We cannot allow this process to be hijacked by efforts to drum up criminal investigations of either candidate,” Mr. Barr said. “I am committed that this election will be conducted without this kind of interference.”

Corrections & Amplifications

Write to Dustin Volz at dustin.volz@wsj.com