Legal experts say the president still has authority to fire Mueller, who was appointed to investigate Russia's meddling in last year's presidential election, even though the special-counsel statute that existed when the Watergate cover-up was being investigated was changed after the long investigation of the Clinton presidency.
PBS' Judy Woodruff reported Monday that Trump friend Chris Ruddy told her after visiting the White House that Trump is "considering perhaps terminating" Mueller - a decision that would set off a massive political backlash.
Mr Trump does not have the authority to dismiss Mr Mueller, and he would have to ask Mr Rosenstein, who appointed the special counsel, to do so.
Collins' question wasn't the only time the Mueller investigation came up during Rosenstein's hearing on Capitol Hill.
Susan Collins, R-Maine, asked Rosenstein during an open Senate Appropriations subcommittee hearing what he would do if Trump asked him to fire Mueller. "If there were not good cause, it wouldn't matter to me what anybody says", he testifies.
"If there were good cause, I would consider it".
'He apparently couldn't find a single pro-Trump attorney to hire and I just think that that's a rigged game.
If Trump orders Rosenstein to fire Mueller and Rosenstein complies, the position of a special counsel would remain vacant, until - or unless - Rosenstein appoints another one.
Trump lashes out over probe into possible obstruction of justice
The President began his day by saying there was "zero proof" of any links between Russian Federation and members of his campaign. Trump told Comey that Flynn had done nothing wrong, the memo said.
Sessions said he recused himself from the Justice Department's Russian Federation investigation only because of a regulation to require the step because of his involvement in the Trump campaign. But Sessions has recused himself from the investigation, which means if Trump wants Mueller fired, Deputy Attorney General Rod Rosenstein would have to do it.
Rosenstein said "no, I have not", when asked by Sen. He later noted that it was up to Mueller to determine whether the issue might be part of his investigation, and he thought Mueller "ought to review that". The fallout from the James Comey firing was mitigated, at least to some degree, by the acceptance of Mueller to take over the investigation. "I'm not going to follow any orders unless I believe those are lawful and appropriate orders", he said, emphasizing that the attorney general "actually does not know what we're investigating". Brian Schatz, D-Hawaii.
Comey testified to the Senate Intelligence Committee last week that he gave one of his associates, a Columbia professor, permission to share information from memos Comey wrote about his private conversations with Trump. That hearing will be public and is scheduled to begin at 2:30 p.m.
"I think he's weighing that option, " Ruddy said in an interview with the PBS NewsHour.
Mr. Trump, Ruddy said, "was looking at [Mueller] potentially to become the next Federal Bureau of Investigation director".
Whether Trump can tolerate a free-ranging investigation, directed by Mueller, that could raise questions about the legitimacy of his Electoral College victory, the topic that most sparks his rage, will be a critical test for a president who has continued on Twitter and elsewhere to flout the advice of his staff, friends, and legal team. Ruddy never spoke to the president regarding this issue.
Sessions, a close campaign adviser to Trump and the first senator to endorse him, stepped aside from the investigation in early March after acknowledging he had spoken twice in the months before the election with the Russian ambassador.
ROSENSTEIN: Well, no, because it's not necessary, because the department is a hierarchy, and so nothing gets to the attorney general about matters he's recused from unless they come through my office. Gingrich, author of a new book called "Understanding Trump", added in an interview: "I think Congress should now intervene and they should abolish the independent counsel".