Washington (CNN) Months after his reported effort to fire special counsel Robert Mueller, President Donald Trump is still fuming over the Russia investigation and has Deputy Attorney General Rod Rosenstein in his crosshairs.

The President has been venting about Rosenstein -- who oversees Mueller and the special counsel investigation -- in recent weeks, according to four sources familiar with the situation. At times, Trump even gripes about wanting Rosenstein removed, two of those sources said. One source said the President makes comments like "let's fire him, let's get rid of him" before his advisers convince him it's an ill-fated idea.

Trump's first year in office has been marked by his preoccupation with an investigation he has dubbed a "witch hunt" and his reported efforts to bring it to a swift conclusion.

After Attorney General Jeff Sessions recused himself from the investigation, Trump took the dramatic step of firing FBI Director James Comey -- but only after asking him for a pledge of loyalty and to back off the FBI investigation into former national security adviser Michael Flynn, according to Comey.

The move led to the appointment of a special counsel in May 2017, a development that continues to infuriate the President. While Trump often directed that anger at Sessions, by June he turned his ire to Mueller and wanted the special counsel fired, a source confirmed to CNN after it was first reported in The New York Times. White House counsel Don McGahn refused to order the Justice Department to fire Mueller, the source said, because he disagreed with the President's reasoning.

Trump appeared in good spirits Friday morning as he prepared to deliver remarks in Davos, a source who spent time with Trump behind the scenes said. There was no sign that he was perturbed by reports about his move to fire Mueller, this person said, and Trump didn't bring up the topic.

But ever since a special counsel was named, the President's frustrations have continued to simmer. Trump has come to view Rosenstein as one-in-the-same as Mueller -- another government official who is out to get the President, one source said.

Sources described Trump's frustration with Rosenstein as largely bluster.

"When this comes up -- everyone says, 'That's the death march. That's not going to accomplish anything,'" said one source familiar with the situation.

Asked for comment, White House special counsel Ty Cobb said in a statement, "We do not find it to be a coincidence that there is an onslaught of false stories circulating in what appears to be a coordinated effort to distract and deflect from new revelations about already reported bias and corruption. We continue to cooperate with the Special counsel and out of respect for that process will not weigh in further."

The Department of Justice declined to comment.

Mueller's office has zeroed in on decisions that seemed to dominate so much of the President's thinking in his first year in office: the firings of Flynn and Comey, as well as any pressure he may have exerted on Sessions.

The President's motives in these situations aren't always entirely transparent.

One source said that, back in June, the President and his attorneys had multiple discussions about whether they should make an issue of Mueller's conflicts. The source disputed that Trump had ordered Mueller to be fired.

In explaining his urge to fire Mueller, The Times, citing two people, said the President said there was a conflict of interest based on Mueller's dispute with one of his golf clubs over fees, the fact that Mueller worked at the same law firm that was representing Trump's son-in-law Jared Kushner, and that Mueller interviewed for the FBI director position the day before he was named special counsel. A source tells CNN another reason Trump wanted to fire Mueller was his perception that Mueller was close friends with Comey. The two men are professional acquaintances.

As for McGahn, he was already prone to becoming frustrated with the President and had threatened to quit over different issues, including Trump's choice of lawyers to defend the President and White House, according to a source familiar with the situation. But the pressure in June to fire Mueller caused McGahn to tell others he would resign, and this time he packed up some of his belongings in boxes, according to two people familiar with the matter.

Before the news broke Thursday, those who have worked closely with McGahn in recent weeks said they believe he's committed to staying at the White House. But his fate could depend on how Trump reacts upon his return to Washington.

Next week, the special counsel's team is set to question former White House chief strategist Steve Bannon about the Flynn and Comey firings, people familiar with the investigation said.

It was a topic that several White House officials have been asked about. The questions "covered the waterfront" ranging from the campaign, connections to Russians, and the meeting at Trump's Bedminster, New Jersey, golf club where the decision to fire Comey was made, one person familiar with one witness' testimony said.

"They're really digging into the Comey obstruction piece," this person said.

Sources familiar with the matter told CNN that Mueller had provided Trump's lawyers with possible interview topics including Trump allegedly asking Comey to drop an investigation into Flynn and Trump's outreach to leaders of the intelligence community about the Russia investigation.

The new information about Trump's desire to fire Mueller -- which the President deemed "fake news" on Friday -- puts his team on a collision course with the special counsel as the two are in talks about the President being interviewed.

Trump told reporters Wednesday he was "looking forward" to an interview with Mueller and would do it under oath. "There has been no collusion whatsoever. There's no obstruction whatsoever," he said. He mockingly told reporters, "You fight back, oh, it's obstruction."

White House lawyers quickly walked that back with one attorney, Cobb, saying the President "spoke hurriedly and intended only to say that he was willing to meet."

John Dowd, the President's personal lawyer, struck a stronger note Thursday, saying, "I will make the decision on whether the President talks to the special counsel. I have not made any decision yet."

It's unclear how the latest revelations will play out politically and legally. If Trump's legal team decides to not agree to an interview, Mueller's team could subpoena his testimony before a grand jury, which could set up a court battle.