If President Donald Trump is truly thinking about firing special counsel Robert Mueller, he should reconsider and "don't waste our time," the top Democrat on the House Intelligence Committee said Monday.

Rumors that Trump was considering firing Mueller — whom the Justice Department appointed to investigate allegations that Trump's presidential campaign may have coordinated with Russia — emerged Monday afternoon in a Washington version of the game Telephone.

The idea gained traction after Christopher Ruddy, chief executive of the conservative news site and TV network Newsmax, said of Trump in an interview on "PBS NewsHour": "I think he's considering perhaps terminating the special counsel. I think he's weighing that option."

How reliable Ruddy's comments are is open to question. He is often described as a close friend of Trump's, and reporters spotted him leaving the White House on Monday. But a source familiar with the visit told NBC News that any meeting Ruddy was to have had with the president was postponed.

Still, his remarks prompted this reply on Twitter from Rep. Adam Schiff of California, the top Democrat on the Intelligence Committee, who said the committee would simply reappoint Mueller and then advised the president: "Don't waste our time."

If President fired Bob Mueller, Congress would immediately re-establish independent counsel and appoint Bob Mueller. Don't waste our time. — Adam Schiff (@RepAdamSchiff) June 12, 2017

"I can't imagine that they're going to be crazy enough to go through with this threat," Schiff said later in an interview on MSNBC's "Hardball," adding: "I think it's just a way of raising doubts about this very good man who's respected by people on both sides of the aisle."

Ruddy himself told PBS: "I personally think it would be a very significant mistake, even though I don't think there's a justification for a special counsel in this case."

House Speaker Paul Ryan on Tuesday morning refused to "get into hypotheticals," but said he would advise Trump to keep Mueller on board.

"I think we should let Bob Mueller do his work and get to the bottom of it and get to the bottom of it quickly so that he can be vindicated," Ryan said on the conservative radio program "The Hugh Hewitt Show." "Let's not forget what this originally is all about: Russia is up to no good. Russia is trying to meddle into our elections."

Before the rumor of a Russia investigation shakeup took on a life of its own, other prominent Republicans had been questioning Mueller's mandate and his fitness to carry it out.

In a panel discussion over the weekend on "Fox News Sunday," former House Speaker Newt Gingrich, a close unofficial adviser to Trump, called Mueller's investigation "a witch hunt."

Gingrich noted the congressional testimony last week of James Comey, whom Trump fired May 8 as director of the FBI, who acknowledged that he had leaked his own memos of his conversations with Trump to force the appointment of an independent counsel.

"You have a director of the FBI deliberately leaking in order to create a special counsel who we're now supposed to believe is going to be this neutral figure," Gingrich said Sunday. "I think that's just nonsense."

But Kenneth Starr, who investigated the Whitewater and Monica Lewinsky scandals as an independent counsel during the administration of former President Bill Clinton, rallied to Mueller's defense.

"Bob Mueller has the integrity to call them as he sees them," Starr said Monday in an interview on ABC's "Good Morning America" before Ruddy spoke. "You have to look at everything."

Even Jay Sekulow, a member of Trump's legal team, said, "I can't imagine that that issue is going to arise," when asked about the possibility of firing Mueller on the same program Sunday.

Another complication is that under the independent counsel statute, Trump can't fire Mueller. Short of impeachment, only the attorney general — in this case, Acting Attorney General Rod Rosenstein — can do so. And it was Rosenstein who appointed Mueller in the first place.

However, Trump could follow the example of Richard Nixon, who eventually got rid of Archibald Cox, the special prosecutor during the Watergate scandal, in October 1973. The turn of events was known as the Saturday Night Massacre.

Nixon directed Attorney General Elliot Richardson to fire Cox, but Richardson resigned rather than comply. So Nixon turned to Deputy Attorney General William Ruckelshaus — who also resigned.

Eventually, Solicitor General Robert Bork, abruptly promoted to acting attorney general by his superiors' resignations, carried out Nixon's order.