With the ex-FBI director's investigation making inroads into the president's inner circle in recent weeks, Republicans have cast doubt on its impartiality and pressed for a new independent prosecutor to investigate anti-Trump bias.
This in turn has led to speculation that the administration could be laying the groundwork for firing Mueller.
In a letter to congressional committees, Trump campaign lawyer Kory Langhofer alleged the General Services Administration (GSA) "unlawfully produced" private materials, including privileged communications which Mueller then used as part of his Russia probe.
GSA is the government agency which supports presidential transitions -- the interim period after a new president is elected but before he takes office.
Langhofer wrote that Mueller's office "received from the GSA tens of thousands of emails, including a very significant volume of privileged material," according to a copy of the letter published by Politico.
The letter added that a warrant should have been obtained for such materials, and said it violated the Presidential Transition Act.
A spokesman for Mueller, Peter Carr, responded on Sunday to the accusations to CNN.
"When we have obtained emails in the course of our ongoing criminal investigation, we have secured either the account owner's consent or appropriate criminal process," CNN quoted him as saying.
Probe tightens
Two administration officials meanwhile Sunday sought to downplay speculation that Trump was preparing to fire Mueller, who was appointed after Trump abruptly fired then FBI director James Comey, who was leading an agency probe into the Russia affair.
Treasury Secretary Steven Mnuchin told CNN: "I haven't heard anything about any firing, but we have to get past this investigation. It's a giant distraction."
Pressed further, he added: "I don't have any reason to think the president is going to do that, but that's obviously up to him."
Marc Short, Trump's director of legislative affairs, was asked on NBC if Trump was laying the groundwork for firing Mueller and responded: "There's no conversation about that whatsoever in the White House."
Critics of the president meanwhile have reacted with dismay and sought to characterize the allegations of illegality as an attempt to stymie the probe, which has already led to charges against Trump's former national security advisor Michael Flynn and three other people linked to the presidential campaign.
"This is another attempt to discredit Mueller as his #TrumpRussia probe tightens," tweeted Democratic Congressman Eric Swalwell.
Republicans asked the US Justice Department on Wednesday to name a new independent prosecutor to probe alleged bias in the FBI after a top agent's anti-Trump text messages were released to the media.
The special counsel's critics have seized upon the case of Peter Strzok, a senior FBI agent whom Mueller removed from his team for sending text messages critical of Trump.
In addition to whether Trump's campaign team colluded with Russia to help him win last year's election, the special counsel is also looking at possible obstruction of justice linked to Trump's firing of Comey.
Trump did so in May and later acknowledged he had the Russia probe in mind when he sacked him.
Comey, for his part, wrote a memo alleging Trump had asked him to drop his investigation into Flynn, an act which some say could constitute obstruction of justice and thus grounds for seeking Trump's impeachment.
