A former aide to political operative and longtime confidant of President Donald Trump, Roger Stone, has been held in contempt for defying an order to testify in special counsel Robert Mueller's Russia probe.

That aide, Andrew Miller, lost a lawsuit earlier this month in U.S. District Court challenging Mueller's subpoena requiring him to appear before a grand jury.

Miller's lawyer, Paul Kamenar, told CNBC in a phone call that Miller refused to comply with the subpoena "in order to be held in contempt and appeal the judge's underlying decision" rejecting his lawsuit.

Outside a Washington federal courthouse on Friday, Miller's attorneys said that they are intentionally challenging the legitimacy of the Russia investigation, specifically regarding Mueller's appointment.

They said the case is now headed to the Court of Appeals, and they hope it will reach the Supreme Court, NBC News reported.

Carl Tobias, a law professor at the University of Richmond, said he doesn't think Miller is likely to find a welcome greeting in the D.C. appeals court. "They are used to resolving, as is [U.S. District Court Judge Beryl] Howell, these kind of disputes," he said, arguing that the Supreme Court has spoken to these narrow issues of authority "pretty clearly."

"I think he's swimming upstream," Tobias said of Miller.

The special counsel's office declined to comment.

Miller is one of a handful of Stone's associates who have been targeted in Mueller's investigation of Russian meddling in the 2016 U.S. presidential election.

Kristin Davis, the former "Manhattan Madam" who worked with Stone and Miller in a 2010 New York gubernatorial campaign, told NBC News she would appear before the special counsel's grand jury on Friday. Davis, who was subpoenaed last month, has said "I don't have anything" to offer regarding Russian collusion, but has agreed to cooperate with Mueller's requests.

CNBC previously reported that Mueller looking into Stone's ties to ex-Trump campaign advisor Rick Gates, who struck a plea deal with Mueller back in February and agreed to fully cooperate with the probe.

Stone's lawyer, Robert Buschel, told CNBC at the time that "Roger Stone did not have any substantive or meaningful interaction with Rick Gates during or leading up to the 2016 campaign."