Reports surfaced over the weekend alleging that Trump has repeatedly complained about Rosenstein's role in the investigation, and that some Republicans have begun to question the deputy attorney general's impartiality.
On Friday, CNN reported that Trump has recently focused his ire on Rosenstein, who appointed special counsel Robert Mueller and was the author of a memo initially used by the White House to justify firing former FBI Director James Comey. Trump, sources familiar with the situation told CNN, had asked to "get rid of" Rosenstein before being dissuaded by his advisors.
Later, The New York Times reported new leaks from a still-classified memo in Congress promulgated by House Intelligence Committee Chairman Devin Nunes. The memo reportedly singles out Rosenstein for extending a surveillance warrant on former Trump campaign advisor Carter Page without adequate cause.
Those reports followed the bombshell news that Trump had attempted to fire Mueller in June 2017, shortly after Rosenstein appointed the former FBI director to take over the Russia investigation.
Neither the White House nor the Justice Department immediately responded to CNBC's requests for comment.
Without the memo itself, or the underlying FISA application documents that reportedly provided its source material, it is difficult to interpret the reported claims, experts told CNBC.
"One cannot tell if there's any merit to it without knowing what the basis is," said Daniel Feldman, a law professor at John Jay College of Criminal Justice. Still, the notion that Rosenstein exceeded his authority in extending the surveillance warrant on Page, even if the Steele dossier was used as justification, rang hollow to Feldman.
"An agency's investigative discretion is wide," he said. With few exceptions, "as long as the investigation is within the mission of the agency, then courts do not second-guess the normal exercise of discretion in how the agency is conducting the investigation."
The Times, citing three sources familiar with the matter, reported that the memo framed the surveillance extension as partly relying on research from Fusion GPS investigator Christopher Steele, a former British spy and the author of the now-infamous dossier alleging salacious connections between Trump and Russia.
But whether Rosenstein achieved the warrant fairly or not is irrelevant to the personal politics involved, others said.
"He wants to fire Rosenstein because he's a thug," Sol Wisenberg, a leading white-collar attorney who was deputy independent counsel in the Whitewater investigation, said of Trump.
The lack of transparency around documents that could potentially show improper use of the Steele dossier might be justifiably confounding for the president, Wisenberg said. "But there's no no mystery about the fact that he's livid at [Attorney General Jeff] Sessions for recusing himself and livid at Rosenstein for appointing a special counsel."
It could also be the case, multiple experts said, that Trump hopes firing the deputy attorney general will allow him to appoint someone more amenable in Rosenstein's place.
Rosenstein has been an occasional target of Trump's criticism since Mueller's appointment in May 2017.
On Monday, Bloomberg reported that Trump became angry on his flight to the World Economic Forum in Davos, Switzerland, after learning that a Justice Department official sent a letter warning against a public release of the House Republicans' memo on alleged FISA abuses. Trump considered the letter another example of the Justice Department's political motivations, sources told Bloomberg.
Justice Department officials, the memo reportedly says, weren't clear enough when applying for the warrant that their information came, in part, from the Steele dossier.
Daniel Farber, a professor of constitutional law at Berkeley Law School, said that misrepresenting the warrant application would be one of the only ways Rosenstein's Justice Department could be accused of misconduct in this scenario.
"Let's say he's relying on material but he attributes it to the CIA rather than an outside source. He would be intentionally misrepresenting facts," Farber said, though he considered it unlikely. "I would be surprised," he said.
Trump and some Republicans have decried the memo, parts of which are reportedly unverified, as being a politically motivated fabrication.
While the right-leaning Washington Free Beacon said it was the first to commission Fusion GPS to investigate Republican primary candidates in 2016, Steele was later paid by the Democratic National Committee, the presidential campaign of former Secretary of State Hillary Clinton and the FBI.
On Monday, White House press secretary Sarah Huckabee Sanders denied that Trump had lost confidence in Rosenstein:
"When the president no longer has confidence in someone, you'll know."