The associate, Lev Parnas, told a representative of the incoming government that it had to announce an investigation into Mr. Trump’s political rival, Joseph R. Biden Jr., and his son, or else Vice President Mike Pence would not attend the swearing-in of the new president, and the United States would freeze aid, the lawyer said.

The claim by Mr. Parnas, who is preparing to share his account with impeachment investigators, challenges the narrative of events from Mr. Trump and Ukrainian officials that is at the core of the congressional inquiry. It also directly links Mr. Giuliani, the president’s personal lawyer, to threats of repercussions made to the Ukrainians, something he has strenuously denied.

But Mr. Parnas’s account, while potentially significant, is being contradicted on several fronts. None of the people involved dispute that the meeting occurred, but Mr. Parnas stands alone in saying the intention was to present an ultimatum to the Ukrainian leadership.