Mr. Parnas’s lawyer, Joseph A. Bondy, confirmed that account and added that his client had met with Mr. Firtash at Mr. Giuliani’s direction and encouraged the oligarch to help in the hunt for compromising information “as part of any potential resolution to his extradition matter.”

Mr. Firtash’s relationship to the Trump-allied lawyers — Victoria Toensing and Joseph diGenova — has led to intense speculation that he is, at least indirectly, helping to finance Mr. Giuliani’s campaign. But until now he has stayed silent, and many of the details of how and why he came to hire the lawyers have remained murky.

In the interview, Mr. Firtash said he had no information about the Bidens and had not financed the search for it. “Without my will and desire,” he said, “I was sucked into this internal U.S. fight.” But to help his legal case, he said, he had paid his new lawyers $1.2 million to date, with a portion set aside as something of a referral fee for Mr. Parnas.