Republicans cheered the news that the Justice Department has named a new investigator to probe the origins of the FBI’s anti-Trump investigations in 2016, even as the scope of the probe expands to include intelligence agencies.

Attorney General William P. Barr, who on Monday revealed he’d named Connecticut U.S. Attorney John Durham to probe the premise of the Trump investigations, has also roped in CIA Director Gina Haspel and Director of National Intelligence Dan Coats to help with the probe of their departments, The Associated Press reported.

“We wanted this a long time ago,” Rep. Jim Jordan, Ohio Republican, told The Washington Times. “I have been so impressed with the attorney general and his commitment to putting a team together and getting to the bottom of this.”

The exact outlines of Mr. Durham’s investigation remain unclear, and it’s also not immediately apparent what it means for John Huber, the U.S. attorney in Utah whom then-Attorney General Jeff Sessions had named in 2017 to look at some of the same ground.

Mr. Jordan described it as a change of pitcher in order to get things going.

“Sometimes the manager has to make a trip to the mound and make a change,” Mr. Jordan said. ‘We haven’t heard anything from John Huber.”

Indeed, it was Republicans’ frustration with Mr. Huber’s lack of public progress that fueled calls for a new special counsel to get to the bottom of the Obama administration’s efforts to snare the Trump campaign.

Mr. Barr, rather than name a special counsel, told lawmakers he’ll oversee the review himself, and now he has a point man to gather the information.

Mr. Durham’s probe will be the third examining the actions of the FBI and Justice Department related to the 2016 probe. Justice Department Inspector General Michael E. Horowitz has commenced an investigation and is expected to release a report this month or next.

As a U.S. attorney, Mr. Durham can extend his probe beyond the limits of the inspector general, whose power is constrained to only current government employees.

President Trump, while denying he had a hand in the move, said he was glad to see it.

“I am so proud of our attorney general, that he is looking into it,” the president told reporters Tuesday. “I think it’s great.”

The president had been among those agitating for a look at how the Obama administration treated him during the campaign. Mr. Trump’s calls grew stronger after special counsel Robert Mueller concluded his investigation revealing there was no evidence of conspiracy between the campaign and Russia to subvert the election — the charge that the FBI had been pursuing.

Mr. Durham brings a sterling reputation to his task.

He’s also had bipartisan backing in the past, including his 2017 nomination to serve as U.S. attorney in Connecticut, where the state’s two Democratic senators recommended him and praised him as a “fierce and fair prosecutor.”

He led numerous government corruption probes for attorneys general from both parties, including exposing FBI agents who’d been turned by mobster James “Whitey” Bulger. Both the Bush and Obama administrations, meanwhile, asked him to lead probes into matters dealing with CIA interrogation of terrorism suspects.

Mr. Durham has already quietly been working for months on a related inquiry into politically motivated leaks, and the Hartford Courant reported he’s built a team of investigators, giving him a head start on the probe into 2016 decision-making by the Obama administration.

Republicans say they see plenty of fishy activity to investigate, including the FBI deploying informants to try to get close to the campaign and using a now-discredited Democratic opposition research document, the Steele dossier, to help justify a secret surveillance warrant on a Trump campaign adviser.

Mr. Barr has called those activities spying — though he says it remains to be seen whether the spying was justified.

Senate Judiciary Chairman Lindsey Graham, who had been agitating for a special counsel and had vowed his own probe, said Tuesday he was going to step back and let Mr. Durham go at it.

“We finally have somebody outside of politics,” he said. “I don’t want to get in their way, I don’t want to mess up his criminal investigation, and I don’t want to put people at risk, so I’m going to back off.”

One Democrat called the probe “a total distraction, saying the FBI was correctly investigating whether Russia was interfering with the 2016 election. Mr. Mueller uncovered plenty of evidence that Russia sought to manipulate the election, according to the 448-page redacted report.

“The investigation had nothing to do with the [Steele] dossier, it frankly had nothing to do with the Trump campaign. It was on actionable intelligence from foreign ambassadors concerned that our democracy was under threat,” Rep. Ro Khanna of California told CNN.

Sol Wisenberg, a former deputy on Kenneth Starr’s Whitewater investigation, said Mr. Durham’s background demonstrates the importance Mr. Barr is placing on this investigation.

“It shows me that Barr is really serious,” he said.

As a U.S. attorney, Mr. Durham will have access to subpoena powers and a grand jury, just as a special counsel would.

But those were the same selling points Mr. Sessions’s team used last year to justify putting Mr. Huber on the case.

The two probes on the surface appear very similar. Mr. Huber was to review some of the early stages of the Russia probe, including a surveillance warrant the FBI obtained to watch Trump campaign figure Carter Page.

“It seems to me it is hard to separate the two of them,” Mr. Wisenberg said.

Mr. Durham’s probe does appear to be more expansive on one front, with the review extending beyond the FBI and Justice Department to include intelligence activities.