(CNN) Supreme Court nominee Brett Kavanaugh has privately told senators he views the appointment of a special counsel by the Justice Department as appropriate, a comment that could shed new light about his views of Robert Mueller's investigation into Donald Trump's presidential campaign, according to sources familiar with the meetings.

But Kavanaugh has also stood by his stated views that question whether a sitting US president can be indicted on criminal charges, instead saying Congress should play the lead role in impeaching and removing a president -- and also enact a law ensuring a president can be indicted after leaving office.

The comments, which multiple sources said were relayed to senators as he's made the rounds on Capitol Hill, give a glimpse into how Kavanaugh is carefully handling questions about his views on executive power at a time when Trump and his associates face growing legal pressure from the Mueller probe. The sources say the nominee is careful not to tip his hand on his views of the Mueller investigation's constitutionality, given that he could rule on matters stemming from the probe, leaving ample questions about his views.

Indeed, if confirmed to the Supreme Court, Kavanaugh's views could carry major consequences -- particularly as Trump's legal team has warned it would fight a potential special counsel subpoena calling for the President's testimony. Senators tell CNN Kavanaugh has not specified his views on a presidential subpoena.

And legal experts said this is still an open question: whether Kavanaugh believes regulations to protect a special counsel from being fired without 'good cause' are indeed constitutional.

Sen. Jeff Flake, a GOP member of the Senate Judiciary Committee who has been a fierce Trump critic and a defender of Mueller, discussed with Kavanaugh his career working with the independent counsel Ken Starr, who investigated the Clinton White House. Flake later told CNN that he doesn't believe that Kavanaugh would view Mueller as unconstitutional.

After noting some senators have expressed concerns that Kavanaugh would be skeptical about the legality of the Mueller probe, Flake said: "I don't share those concerns."

There are differences between a special counsel like Mueller and an independent counsel like Starr. Mueller, who was appointed last year by Deputy Attorney General Rod Rosenstein to investigate Russian interference in the 2016 elections and possible collusion with the Trump campaign, works directly for the Justice Department. The law that established the independent counsel, which operates more freely, expired in 1999 when it was replaced by the more modest Justice Department regulation governing special counsels.

After working for Starr from 1994-1998, Kavanaugh has been a sharp critic of independent counsels -- and has said that the Supreme Court should overturn the 1988 ruling, Morrison v. Olson, that upheld the constitutionality of the independent counsels. He has said publicly that it's the role of Congress -- not an outside entity -- to investigate a president.

And he has repeatedly been skeptical about whether a sitting president can be indicted, including in a much-discussed 2009 Minnesota Law Review article where he asserted that a President "should be excused from some of the burdens of ordinary citizenship while serving in office." Instead, he said, Congress could enact a new law extending the statute of limitations so a president could face charges once leaving office.

In a private meeting with North Carolina Republican Sen. Thom Tillis, Kavanaugh reiterated his past statements questioning whether a "sitting president" can be indicted, instead detailing the constitutional role of Congress to take action against a president. In meetings with senators, Kavanaugh points out those views also are in line with long-standing Justice Department policy, saying a sitting president can't be indicted. But he has been careful -- and has stopped short of asserting it's unconstitutional to indict a sitting president, given that the Supreme Court has not ruled on that matter, according to one source briefed on the matter.

But Tillis said he was comfortable with Kavanaugh's explanation laying out his views about the separation of powers: that the House impeaches a president, followed by the Senate convicting a president and removing him from office. At that point, Kavanaugh argued, it would be the proper time to carry on with any criminal prosecution for a president after leaving office.

"What he was saying is there are devices to remove a president at such time that you could carry on prosecution," Tillis said. "So he was talking more about process."

Tillis added: "I thought it made perfect sense. You'd have to really distort his words not to come to the same conclusion."

While Tillis and Kavanuaugh did not discuss the Mueller probe explicitly and how he would view a subpoena compelling Trump to testify before a federal grand jury: "It's probably a question that'll come up in the hearings," Tillis added.

Stephen Vladeck, a law professor at the University of Texas and CNN legal analyst, said that Kavanaugh's private comments about a special counsel still leave open some key questions about his views.

"Does Mueller therefore have the power to enforce a subpoena against Trump?" Vladeck said of the questions that Kavanaugh's comments raise going forward. "Is the provision protecting Mueller from being fired without good cause also constitutional?"

Kavanaugh has written, at least briefly, about special counsels in the recent past. In an opinion from January for a federal appeals court in DC, after Mueller was appointed, Kavanaugh did acknowledge a difference between an independent counsel and special counsel. He did not explicitly discuss his view about the legality of a special counsel's investigation of a president.

"The independent counsel is, of course, distinct from the traditional special counsels who are appointed by the Attorney General for particular matters," Kavanaugh wrote. "Those special counsels ordinarily report to and are removable by the Attorney General or the Deputy Attorney General."

Kavanaugh has been making the rounds on Capitol Hill, meeting with 39 Republican senators where he's discussed his career working with Starr and at the George W. Bush White House before becoming confirmed for a seat on the US Court of Appeals for the District of Columbia Circuit in 2006. He has only met with one Democratic senator: West Virginia Sen. Joe Manchin, a moderate in a difficult re-election race.

Manchin would not discuss the details of his closed-door, two-hour Monday meeting with Kavanaugh, saying, "I don't know if there's any topic that I missed." But he said that Kavanaugh was forthcoming and stressed that he would be an independent-minded jurist.

"When the hearing is over, I will want to call him back in to sit down and go over basically what we heard today," Manchin said Monday.

Sen. Susan Collins of Maine, a moderate Republican and a key swing vote, said she wants to hear Kavanaugh's views of the Mueller probe when they meet, likely before his confirmation hearings, which have not yet been set. On Monday, Collins endorsed her party's strategy to limit the request of documents to vet for Kavanaugh, saying there would still be "more than 1 million" pages of records to review of his past -- even though the Senate Judiciary Committee is not asking for documents related to his time from 2003-2006 serving as Bush's staff secretary, enraging Senate Democrats.

Still, Sen. Lisa Murkowski of Alaska, another GOP swing vote, did not express concerns Tuesday when asked if she believed records from Kavanaugh's time as staff secretary should be turned over to senators as well.

"We are at about 1-million-plus (pages) right now," she told CNN.