There is no evidence of wrongdoing in Ukraine by either Joe Biden or his son Hunter, who served on the board of a Ukrainian gas company.
Giuliani rebutted Parnas' claim, telling the Times, "Categorically, I did not tell him to say that."
A person familiar with the events told the newspaper the meeting consisted of Parnas, Fruman and Serhiy Shefir, a close aide to Zelensky, drinking coffee at an outdoor cafe days before Zelensky's inauguration.
Fruman, who also attended the meeting, pushed back on Parnas' account. Fruman's lawyer John Dowd told the Times that the men looked only to meet with Zelensky and never brought up the US aid or Pence's presence at the inauguration.
CNN has reached out to Giuliani and representatives for Parnas and Fruman for comment.
Shefir told the Times in a statement that he met with Parnas and Fruman. He noted that Parnas and Fruman did not bring up military aid and had asked for a meeting between Zelensky and Giuliani.
"We did not treat Mr. Parnas and Mr. Fruman as official representatives, and therefore we did not consider that they could speak on behalf of the U.S. government," Shefir said in a statement to the Times.
CNN's Marshall Cohen and Kevin Bohn contributed to this report.