Burkman has frequently called on President Donald Trump to fire Mueller, along with Deputy Attorney General Rod Rosenstein, who oversees the special counsel, and Attorney General Jeff Sessions, who had recused himself from interfering in the probe. He even offered a $25,000 reward to any whistleblower who could offer concrete evidence of FBI wrongdoing in those investigations, The Washington Times reported in February.
The special counsel's office confirmed to CNBC that it learned about the "scheme" from journalists who had been approached by a woman alleging that she had been offered $20,000 by Burkman "to make accusations of sexual misconduct and workplace harassment against Robert Mueller."
Burkman did not immediately respond to a request for comment.
In a Facebook video also posted Tuesday, Burkman claimed "we will unveil the first of the sex assault victims of Robert Mueller." In the video, Burkman said the "first" alleged accuser is "coming out this Thursday at high noon."
He added: "Robert Mueller is a bad guy not just because of what he does inside the courtroom, but because of what he does outside the courtroom."
DOJ spokeswoman Sarah Isgur Flores referred inquiries back to the special counsel. The FBI declined to comment on Carr's statement.