Gingrich, who only a few weeks ago called Mueller a "superb choice" with an "impeccable" reputation, tweeted on Monday: "Republicans are delusional if they think the special counsel is going to be fair".
The White House did not immediately respond to questions about Mr Ruddy's claims.
Mueller was appointed by Deputy Attorney General Rod Rosenstein.
It was Mr Rosenstein who appointed Mr Mueller to the job in the first place, since he took carriage of the Russia investigation from the Attorney-General Jeff Sessions, who was forced to recuse himself because of his own contacts with Russian officials in the lead-up to last year's United States election.
Noted Trump surrogate Newt Gingrich, a former Speaker of the House, reversed himself this week and began attacking Mueller as a vicious partisan whose hiring of Democratic loyalists as deputies has cast his impartiality into doubt.
Of course, Rosenstein's answer does not confirm that Mueller is definitely exploring the issue.
Justice Department regulations stipulate that a special counsel can be fired only for "misconduct, dereliction of duty, incapacity, conflict of interest, or for other good cause", but there may already be such grounds to remove Mr Mueller as special counsel.
The order from the general counsel for the transition team casts a wide net on documents that could shed light on ties between Trump's presidential campaign and representatives of Russia's government.
Trump, angered by reports in Breitbart News and other conservative news media that Mueller was close to James Comey, the FBI director he had fired, repeatedly brought up the political and legal implications of firing someone he viewed as incapable of an impartial investigation.
Harry Kane's equaliser 'huge moment' for England - Gareth Southgate
In the marines, they are life and death situations and we took as much as we could from the game and stood up to be counted. Gareth Southgate's side had taken the lead in the 70th minute when Alex Oxlade-Chamberlain had fired beyond Craig Gordon.
But he could order the Justice Department to do so. 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.
Ultimately, Ryan thinks Trump will heed the warnings and not move to fire the special counsel.
"I am being investigated for firing the FBI Director by the man who told me to fire the FBI Director!"
Mr Gingrich's criticism could be a harbinger of a drive by Trump surrogates to undermine Mr Mueller's authority and, perhaps, lay the groundwork to question his ultimate prosecutorial decisions or seek his removal.
Here's why. According to the rules, technically only the attorney general can fire a special counsel.
The appearances, however, are likely to be overshadowed by Attorney General Jeff Sessions' public appearance at a Senate intelligence committee hearing.
Sessions was initially supposed to testify before the appropriations subcommittee, but requested to send Rosenstein in his place.
President Donald Trump has acknowledged for the first time that he is under federal investigation as part of the expanding probe into Russia's election meddling. He proved as tough as Cox, successfully suing the White House for information, including Oval Office tapes that led to Nixon's resignation nine months later.
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