Mr. Parnas’s account of the meeting, if corroborated, would reveal the earliest known instance of American aid being tied to demands for Ukraine to take actions that could benefit Mr. Trump’s 2020 re-election campaign. It would also represent a more extensive threat — to pull Mr. Pence from the inaugural delegation — than was previously known.

Mr. Trump froze nearly $400 million in military aid to Ukraine shortly before a July 25 call with the country’s president, Volodymyr Zelensky, in which Mr. Trump personally sought investigations into the Bidens and claims that Ukrainians had meddled in the 2016 election. In the call, Mr. Trump did not explicitly link the aid and the investigations.

Mr. Trump has denied a quid pro quo involving aid, and Mr. Zelensky has said he never felt pressured to pursue an investigation.

The meeting in Kiev in May occurred after Mr. Giuliani, with Mr. Parnas’s help, had planned a trip there to urge Mr. Zelensky to pursue the investigations. Mr. Giuliani canceled his trip at the last minute, claiming he was being “set up.”

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Only three people were present at the meeting: Mr. Parnas, Mr. Fruman and Serhiy Shefir, a member of the inner circle of Mr. Zelensky, then the Ukrainian president-elect. The sit-down took place at an outdoor cafe in the days before Mr. Zelensky’s May 20 inauguration, according to a person familiar with the events. The men sipped coffee and spoke in Russian, which is widely spoken in Ukraine, the person said.

Mr. Parnas’s lawyer, Joseph A. Bondy, said the message to the Ukrainians was given at the direction of Mr. Giuliani, whom Mr. Parnas believed was acting under Mr. Trump’s instruction. Mr. Giuliani said he “never authorized such a conversation.”

A lawyer for Mr. Fruman, John M. Dowd, said his client told him the men were seeking only a meeting with Mr. Zelensky, the new president. “There was no mention of any terms, military aid or whatever they are talking about it — it’s false,” said Mr. Dowd, who represents Mr. Fruman along with the lawyer Todd Blanche.