CNN has reported at the time that the White House was given only a brief heads up that Deputy Attorney General Rod Rosenstein was naming Mueller to oversee Russian interference in the 2016 election and potential collusion from Trump's campaign associates.
Mueller is said to be interviewing as early as this week three top intelligence officials as part of the probe, the Post reported, citing "five people briefed on the requests".
Sean Spicer, the White House press secretary, said Ruddy "never spoke to the president regarding this issue".
Under Department of Justice rules, it is actually up to the deputy attorney general to make that call, though the president could bypass that protocol.
Sen. Susan Collins, R-Maine, asked Rosenstein at a budget hearing Tuesday what he would do if Trump ordered him to fire Mueller, who is investigating Russian interference in the USA elections and possible Russian ties to Trump's campaign.
However, White House spokesperson Sarah Huckabee says, "Chris speaks for himself".
Some Trump supporters, including Ruddy and Newt Gingrich, don't think Mueller can be fair because of his close ties with recently fired FBI Director James Comey.
It would be hard to make the case, he said, that the experience of interviewing for Federal Bureau of Investigation director would make it impossible for Mueller to fairly exercise the broad discretion afforded to prosecutors.
Trump ally Dick Morris told Politico on Monday: "I think the idea of having an enemy when you're the object of a special prosecutor is a very important one".
Rosenstein said he would agree to dismiss Mueller only if there were a legitimate basis to do so, and an order from the president would not necessarily qualify. If Rosenstein refused, Trump could fire Rosenstein. "I think the best vindication for the president is to let this investigation go on independently and thoroughly". Mr Mueller's moves, however, indicate he does not share Mr Trump's view. Watching the president fire the special counsel investigating the Russian Federation scandal would test the GOP's tolerance, but it may be wishful thinking to believe Trump's Republican allies have a sense of limits.
New Hampshire Sen. Jeanne Shaheen, a Democrat, opened her line of questioning, in the hearing that was supposed to focus on the Justice Department budget, by asking if Rosenstein had seen any evidence of good cause to fire Mueller. He told MSNBC's "Morning Joe" that "I don't think the Congress would sit still and allow the president to pick his own investigator".