The new line of inquiry reportedly began after revelations that Trump allegedly attempted to persuade former FBI Director James Comey into ending an investigation into Flynn, according to a Comey memo detailing a conversation he had with the president.
The Times does not report what else may have been in the document, which would appear to have been classified.
Even if the Justice Department granted a waiver, the White House would consider using the ethics rule to create doubt about Mueller's ability to do his job fairly, the sources said.
Rosenstein said he first learned of Trump's intent to remove Comey on May 8.
Ric Spooner, a market strategist at CMC Markets, said: "While investors will be relieved that yesterday's selling looks unlikely to be repeated today, it's too early to assume yesterday was a one-day wonder".
Trump's approval rating continues to slip to 41%. Though the story has evolved over the last few days, Mr. Trump has said that "very strong recommendation from the Deputy Attorney General" played a role in his decision to terminate Comey.
In closed-door meetings with lawmakers on Thursday and Friday, Rosenstein said he wrote the memo after Trump told him one day before the May 9 firing that he wanted to dismiss Comey.
Sen. Claire McCaskill (D-Mo.) told reporters afterward that Rosenstein revealed new information about the letter he penned, which Trump cited to justify Comey's firing.
Trump was using a negotiating tactic when he told Russian officials about the "great pressure" he was under, another United States official defended the president to the Times, suggesting that Trump's goal was to create a sense of obligation with Russian officials and to coax concessions on certain issues.More news: Will Smith defends Netflix films at Cannes
When asked about the Justice Department appointing Mueller as Special Counsel, the President deflected the question.
President Trump nominated career Justice Department official Rod Rosenstein to be deputy attorney general in February - but his fate was immediately tied up in the probe of Russian election interference. Officials said it had nothing to do with the Russian Federation investigation.
The White House has struggled since Comey's firing to explain the chain of events that led to it and who exactly made the decision.
"The "great pressure" taken off thing?"
Deputy Attorney General Rod Rosenstein, speaking before lawmakers Friday, defended his controversial memo criticizing former FBI Director James Comey - a document the White House initially used to justify the firing of Comey.
When he announced Mueller's appointment this week, Rosenstein said Mueller will have "all appropriate resources to conduct a thorough and complete investigation".
Lawmakers at both congressional sessions expressed frustration that Rosenstein would say little in answer to their questions about his actions - or others' - before Comey's firing.
Democrats, who have compared the widening scandal to the Watergate break-in and cover-up that brought down Republican former President Richard Nixon in 1974, were quick to pounce on the latest reports. "Director Comey served his country with honor for many years, and he deserves an opportunity to tell his story".
A grand jury in Alexandria, Virginia, recently issued a subpoena for records related to Flynn's business, The Flynn Intel Group, which had been paid more than $500,000 by a company owned by a Turkish American businessman close to top Turkish officials, according to people familiar with the matter.