The plot first came to light through one of the emails referred to the F.B.I., sent to journalists on Oct. 17 by a woman named Lorraine Parsons who said that a man had contacted her with questions about Mr. Mueller because she had worked as a paralegal with him at a law firm in 1974.

She said the man, whom she identified as Bill Christensen and said had a British accent, contacted her a second time and offered her more than $50,000 to “make accusations of sexual misconduct and workplace harassment against Robert Mueller.” To get the money, he said, she would need to sign an affidavit and promised a $10,000 bonus if she did so quickly.

The man added that he was working for Jack Burkman, a Republican lawyer. Ms. Parsons said she declined the financial offer and hung up the phone.

She identified herself to at least two news media outlets who received her email as Lorraine D. Parsons of Fort Myers, Fla., but The New York Times could not find her. Though she wrote in the email that Mr. Mueller was “always very polite to me, and was never inappropriate,” the law firm where Ms. Parsons said she worked alongside him — Pillsbury Winthrop Shaw Pittman, known at the time as Pillsbury, Madison and Sutro — said it had no record of Ms. Parsons working there.

Mr. Burkman is known for peddling right-wing conspiracy theories. He suggested that Russian hit men killed Seth Rich, a 27-year-old Democratic National Committee aide, during the 2016 presidential campaign and that someone tried to kill him while investigating Mr. Rich’s death. Law enforcement officials told Mr. Rich’s parents he was most likely killed during a botched robbery.