Arizona-based attorney Kory Langhofer claims Mueller's acquisition of the emails violated attorney-client privilege and the Fourth Amendment, which prohibits unreasonable searches and seizures.
The e-mails, handed to Mueller by the federal General Services Administration, reportedly number in the "tens of thousands", according to a letter sent to Congress by the organization Trump for America on Saturday.
Langhofer detailed his claims in a seven-page letter sent to Congressional oversight committees over the weekend.
"Additionally, certain portions of the [transition] materials the Special Counsel's Office obtained from the GSA, including materials that are susceptible to privilege claims, have been leaked to the press by unknown persons".
According to Langhofer, the assurance was made by then-GSA General Counsel Richard Beckler, who was hospitalized in August and has since died.
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"I seriously doubt there is anything here to taint the [Mueller] investigation", said William Jeffress, a white-collar defense attorney who represented Vice President Dick Cheney's senior aide, I. Lewis "Scooter" Libby, during the Valerie Plame CIA leak investigation. He did not name the transition officials who used the devices.
Langhofer argues in the letter that decision by GSA officials went against what he calls "the GSA's previous acknowledgement concerning" the Trump campaign's "rightful ownership and control of" transition team materials.
There was plenty of chatter Sunday outside the White House about whether President Trump will fire special counsel Robert Mueller, but apparently nothing of the sort on the inside. (EPA) President Trump being sworn in.
But in his letter Langhofer said that Mueller's team has been "actively using" the emails obtained from GSA as part of the ongoing investigation.
GSA Deputy Counsel Lenny Loewentritt, who Langhofer blames along with other GSA career staff for providing the transition documents to the Federal Bureau of Investigation, told Buzzfeed that Beckler didn't make a commitment to the transition team that requests from law enforcement for materials would be routed through transition lawyers. "Baloney. This is another attempt to discredit Mueller as his #TrumpRussia probe tightens", Rep. Eric Swalwell, a California Democrat and member of the House Judiciary and Intelligence panels, posted on Twitter.
Late Saturday, Mueller's spokesman, Peter Carr, said the special counsel's office has followed the law when it has obtained documents during its investigation.