(CNN) President Donald Trump's pick to take over the Justice Department told Republican senators Wednesday he would not interfere in special counsel Robert Mueller's investigation, according to the incoming chairman of the Senate Judiciary Committee.

Attorney General nominee William Barr was on Capitol Hill meeting privately with senators ahead of his confirmation hearings next week. Following his meeting, Republican Sen. Lindsey Graham of South Carolina told reporters that Barr said he was committed to allowing Mueller to finish his probe.

"I can assure you he has a very high opinion of Mr. Mueller, and he's committed to seeing Mr. Mueller complete his job," Graham said. "I asked Mr. Barr directly, 'Do you think Mr. Mueller is on a witch hunt?' He said no. 'Do you think he'd be fair to the President and the country as a whole?' He said yes."

Barr is meeting with lawmakers on Capitol Hill as he tries to win over senators skeptical of his views on executive power and the special counsel investigation that has driven the agency into a political minefield. One week out from his scheduled confirmation hearings, the stakes of Barr's confirmation fight were raised on the heels of reports that the Justice Department's stalwart No. 2, Rod Rosenstein, is leaving.

Democrats plan to make Barr's views on the Mueller investigation a key question in the hearings — and will demand public assurances similar to what Graham said Barr pledged privately on Wednesday. Democrats have seized on a memo Barr wrote in June outlining a broad vision of presidential authority and concluding that Mueller's inquiry into obstruction of justice was "fatally misconceived." The memo was sent at the time to senior Justice officials and was released as part of a questionnaire Barr submitted to the committee last month for vetting.

Rosenstein himself appointed special counsel Robert Mueller in May 2017, and he maintained day-to-day management of the probe even after Trump installed Matt Whitaker as acting attorney general late last year — a move that replaced former Attorney General Jeff Sessions, who had recused himself from the investigation, and eventually shifted its oversight from Rosenstein to Whitaker. Rosenstein's departure, which is planned for shortly after Barr's potential confirmation, according to a source familiar with the deputy attorney general's thinking, will thrust Barr's views on the Mueller investigation to the center of his confirmation.

If confirmed, Barr would oversee the special counsel's Russia investigation, gaining briefings on its progress and likely the ability to block some investigatory steps before they are taken.

Graham said Barr and Mueller were personal friends with a long history. He said Barr indicated he would not curtail Mueller's investigation, and that Barr would follow the Justice Department's protocols while "erring on the side of transparency" when it comes to sharing Mueller's final report with Congress and the public.

"He had absolutely no indication he was going to tell Bob Mueller what to do or how to do it," Graham said.

Asked about Barr's memo, Graham said the attorney general nominee had concerns about the precedent that could be set over obstruction of justice and the firing of a political appointee.

"That was his opinion about the slippery slope of obstruction of justice charges against a president who wants to terminate a political appointee," Graham said. "He believes that's a slippery slope and I share that view."

Barr, a former attorney general under President George H. W. Bush, met Wednesday with the Graham and Sen. Chuck Grassley of Iowa, the outgoing chairman of the Judiciary Committee. He is expected to meet this week with more senators, including some Democrats.

An old-guard conservative who held some of Washington's most influential legal positions, Barr's nomination last month to succeed Sessions was met with commendation by Justice Department officials and Republicans from across the ideological spectrum. Some Democrats, however, have seized on comments Barr made to newspapers last year criticizing Mueller's team of prosecutors and supporting Trump's calls for investigations into Hillary Clinton.

In a letter last month, Sen. Dianne Feinstein of California, the top Democrat on the committee, sent Barr a list of questions about the origin of the memo, writing, "I read your memorandum with great surprise." She has not yet received a response from Barr, her office said.

Sen. Richard Blumenthal, a Connecticut Democrat, on Tuesday called the memo "deeply worrisome" and said he would seek "an ironclad commitment that he will protect the special counsel from political interference and recuse himself if he refuses to disavow the points that he made in his memorandum."

Barr's comments to Graham on Wednesday did not assuage Blumenthal and other Democrats' concerns.

"I want more than bland ... assurances," Blumenthal said. "I want iron clad, specific commitments, and possibly even recusal."

Similarly, Sen. Sheldon Whitehouse of Rhode Island said he wasn't convinced yet.

"It will be very important to test those assurances through the hearing process," Whitehouse said.

Feinstein said she expects to meet with Barr on Thursday, as is typical ahead of confirmation hearings. But other Democrats on the committee said they did not yet have meetings with Barr on the books after they had requested them, including Whitehouse and Sen. Amy Klobuchar of Minnesota.

"I'm trying to get a meeting ahead of time. I've never not met (with a nominee)," Klobuchar said.

Republicans increased their margin on the judiciary panel to two after their election wins, making it likely that Barr does not need to win the support of any Democrats to advance positively out of the committee after his hearing, two GOP members of the committee repeated their defense of the Mueller probe on Tuesday.

Graham and North Carolina Sen. Thom Tillis were among a bipartisan group of senators that reintroduced legislation that would protect Mueller from "inappropriate removal or political pressure." The bill passed the Judiciary Committee last Congress across party lines but was never brought up before the full Senate for a vote.

Asked about Barr's memo on Mueller, before news of Rosenstein's planned departure broke, Tillis shrugged off Democratic concerns.

"Not yet," he told reporters when asked if he has concerns. "I'll be talking to him before the hearing, and then we'll have the hearing and we'll see where it goes from there."

Other Republicans defended Barr. "He wrote that as a private citizen," Grassley said Tuesday ahead of his meeting. "What you do as a private citizen is one thing. What you do as a public citizen is another."

Next week's confirmation hearing will not be Barr's first before the Judiciary Committee, though it comes after a lengthy hiatus from government service.

As he's prepared, Barr bowed out of plans for an international hunting trip earlier this month, a friend said, and has spent his days studying with a team of DOJ lawyers at the Department of Justice in Washington, according to a Justice Department official.

This story has been updated with additional developments Wednesday.