But intermediaries for Mr. Giuliani worked to organize meetings with people who they believed would have insights into the incoming Zelensky administration and the investigations in which Mr. Giuliani was interested. And in recent days, Mr. Giuliani reached out through intermediaries to request a meeting with Mr. Zelensky, he said, adding, “It’s not confirmed yet.”

If the meeting does occur, Mr. Giuliani said, “I am going to tell him what I know about the people that are surrounding him, and how important it is to do a full, complete and fair investigation.”

He said his efforts in Ukraine have the full support of Mr. Trump. He declined to say specifically whether he had briefed him on the planned meeting with Mr. Zelensky, but added, “He basically knows what I’m doing, sure, as his lawyer.”

The White House did not immediately respond to a request for comment on Thursday.

Mr. Trump has called attention to the scrutiny of Mr. Biden’s son Hunter Biden, and to questions about the former vice president’s involvement in the removal of a Ukrainian prosecutor whose office had authority over investigations of the oligarch whose company paid Hunter Biden.

Mr. Trump has also sought to stir up interest in claims that Ukrainian officials tried to benefit Hillary Clinton in 2016 by focusing attention on Mr. Manafort’s business in Ukraine. The attention forced Mr. Manafort to resign from the Trump campaign, but allies of the Ukrainian officials involved have denied that they acted improperly to benefit Mrs. Clinton’s campaign. Mr. Trump has recently suggested he would like Attorney General William P. Barr to look into the material gathered by the Ukrainian prosecutors.

Mr. Giuliani has been working on the effort with other allies of Mr. Trump whose involvement has not been previously reported, including Victoria Toensing, a lawyer who was named last year, along with her husband, as part of the legal team representing the president in the special counsel’s investigation. The appointment was rescinded less than one week later amid concerns about conflicts of interest, but Mr. Trump’s legal team suggested that Ms. Toensing and her husband, Joseph E. diGenova, would assist the president “in other legal matters.”

On social media and in regular appearances on Fox News, the couple advanced the theory that the special counsel’s investigation was the result of a Justice Department effort to frame Mr. Trump. They increasingly began pushing the claim that “the real collusion began in @Ukraine,” as Ms. Toensing put it in a post on Twitter in March.