But interviews with the two Ukrainian oligarchs — Dmitry Firtash and Ihor Kolomoisky — as well as with several other people with knowledge of Mr. Giuliani’s dealings, point to a new dimension in his exertions on behalf of his client, Mr. Trump. Taken together, they depict a strategy clearly aimed at leveraging information from politically powerful but legally vulnerable foreign citizens.

In the case of Mr. Firtash, an energy tycoon with deep ties to the Kremlin who is facing extradition to the United States on bribery and racketeering charges, one of Mr. Giuliani’s associates has described offering the oligarch help with his Justice Department problems — if Mr. Firtash hired two lawyers who were close to President Trump and were already working with Mr. Giuliani on his dirt-digging mission. Mr. Firtash said the offer was made in late June when he met with Lev Parnas and Igor Fruman, both Soviet-born businessmen involved in Mr. Giuliani’s Ukraine pursuit.

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.