Trump, who isn't typically shy about leveling criticism, especially on Twitter, hasn't commented on the longtime former Federal Bureau of Investigation director.
The morning missive appeared to refer to Rod Rosenstein, the deputy attorney general.
Ruddy, who had been at the White House on Monday, told PBS that Trump is considering terminating the special counsel.
"It makes sense to point out any level of bias, to highlight any biases inherent in Mueller and this probe in order to force him to be transparent", Gingrich said. When The Washington Post reported this week that Mueller was investigating Trump for obstruction of justice, a spokesman for the President's private attorney called the leak "outrageous, inexcusable and illegal".
On Thursday night, he issued a statement lashing out at news stories sourced to anonymous officials and that he believes are causing the President and Republicans to attack the Justice Department, the Federal Bureau of Investigation and Mueller for alleged leaks.
But Rosenstein, too, may ultimately have to hand off oversight of the probe given his own role in Comey's firing.
Robert Mueller, appointed as special counsel in the investigation into Russian interference in the 2016 election, is scrutinizing Jared Kushner's "finances and business dealings, "The Washington Post reported Thursday citing unnamed "officials familiar with the matter". If Rosenstein were out of the picture, responsibility for overseeing Mueller's investigation - and deciding his fate - would fall to Rachel Brand, the department's associate attorney general. In this case, Rosenstein is the acting attorney general, since Attorney General Sessions recused himself.
The former FBI Director said he believed that the president was attempting to get him to drop the investigation into Flynn. "It would be standard practice for the special counsel to examine financial records to look for anything related to Russian Federation".
The officials who described the financial focus of the investigation spoke on condition of anonymity because they were not authorized to speak publicly. Now there's some question of whether President Trump wants to fire Mueller. The order also covers separate inquiries into several key Trump associates including former National Security Adviser Michael Flynn, campaign adviser Paul Manafort, foreign policy aide Carter Page and outside adviser Roger Stone.
The White House has directed questions for details to outside legal counsel, which has not responded. Pence headed the Trump transition until Inauguration Day.
Gingrich and Ruddy said it would be a "mistake" for Trump to remove Mueller, although both said they see the probe as unnecessary.
The probe, which began a year ago as an investigation into Moscow's meddling in the presidential election, has since widened to include prospective money laundering by Trump associates and the possibility that people close to the President were in contact with Russian operatives throughout the campaign. The memo asks for preservation of electronic communications and data, telephone logs, audio recordings, videos, calendars and other items.
The lawyer for the president's son-in-law and close adviser said there was nothing surprising about the Post's report.
United States intelligence agencies concluded in January that Russian Federation interfered in the presidential race to try to help Trump win, in part by hacking and releasing emails damaging to his Democratic opponent, Hillary Clinton.
"They made up a phony collusion with the Russians story, found zero proof, so now they go for obstruction of justice on the phony story", he wrote on Twitter. "Nice", Trump wrote in his first tweet.
And then: "You are witnessing the single greatest WITCH HUNT in American political history - led by some very bad and conflicted people!"