A reconstructed transcript of the call, made public by the White House in October, makes clear that Mr. Trump asked the Ukrainian president to pursue investigations into the Bidens and into one element of his belief that Ukraine worked against his election in 2016: a debunked conspiracy theory that Ukraine rather than Russia was behind the hacking of the Democratic National Committee, and that Ukraine had possession of a server that would shed light on the theory.

“I would like you to do us a favor though,” Mr. Trump said, asking Mr. Zelensky’s government to work with Attorney General William P. Barr and Mr. Giuliani to pursue the investigations.

“I would like you to find out what happened with this whole situation with Ukraine, they say CrowdStrike,” Mr. Trump said, referring to an American cybersecurity firm and the debunked theory about Ukraine’s involvement in the hack of the Democratic Party. “The server, they say Ukraine has it.”

He went on to bring up the Bidens.

“There’s a lot of talk about Biden’s son, that Biden stopped the prosecution, and a lot of people want to find out about that, so whatever you can do with the attorney general would be great,” Mr. Trump said, according to the reconstructed transcript. “Biden went around bragging that he stopped the prosecution, so if you can look into it.”

What Mr. Trump first said in private to Mr. Zelensky, he later said in public. In early October, answering questions from reporters outside the White House, Mr. Trump repeated and expanded on his calls for foreign help in investigating the Bidens.

“I would say that President Zelensky, if it were me, I would recommend that they start an investigation into the Bidens,” Mr. Trump said. “Because nobody has any doubt that they weren’t crooked.”

He also suggested that Ukraine was not the only country that should dig into Hunter Biden’s international business dealings.