Mueller has asked for documents on Trump's reasons for firing former national security advisor Michael Flynn and ex-director of the FBI James Comey, theNew York Times reported on Wednesday.
"The paper quoted Ian Prior, a Justice Department spokesman, as saying: "As the deputy attorney general has said numerous times, if there comes a time when he needs to recuse, he will".
The document requests are the clearest indication yet that at least some aspects of Mueller's probe are focused on the president's behavior in office.
Mueller's investigation is one of many Washington probes delving into alleged election meddling by Russian Federation and potential collusion with the Trump campaign.
A special counsel can be removed by the attorney general or, if the AG has recused himself, the most senior Department of Justice official.
Mueller has requested that the White House turn over all internal communications and documents related to the Federal Bureau of Investigation interview of Flynn in January, days after he took office, as well as any document that discusses Flynn's conversations with Russian Ambassador Sergey Kislyak in December.
"I am convinced that no matter where they end up, this investigation will run to completion even if they fire Mueller", the official said. It was revealed earlier this month that Trump and aide Stephen Miller had drafted, but not sent, an earlier memo that sought to justify Comey's firing.
Rosenstein has said he stands by the memo and his assessment that Comey mishandled the Clinton e-mail investigation by publicly announcing the FBI's findings instead of ceding that authority to the Justice Department.
The fact that Mueller's team would speak with Rosenstein is not surprising given his direct involvement in Trump administration conversations that preceded the May 9 ouster and the evolving White House explanations of it.
Trump met with Russian Foreign Minister Sergei Lavrov on May 10 in the Oval Office - one day after the dismissal. "He was insane, a real nut job", Trump said, according to a document obtained by the Times at the time.
Mueller has also asked for all records related to the June 2016 meeting at Trump Tower with a Russian lawyer hosted by the president's son, as well as all documents related to the White House's response to the publication of that meeting in July 2017.
Former acting Attorney General Sally Yates has said she warned White House counsel Don McGahn in late January that Flynn could be vulnerable to Russian blackmail because he had not been truthful with the White House about his conversations with Kislyak.