WASHINGTON — A senior National Security Council aide on Thursday confirmed a key episode at the center of the impeachment inquiry, testifying that a top diplomat working with President Trump told him that a package of military assistance for Ukraine would not be released until the country committed to investigations the president sought.

In a closed-door deposition, the aide, Timothy Morrison, also said he had been told of a September call between Mr. Trump and the diplomat, Gordon D. Sondland. In that conversation, the president said he was not looking for a quid pro quo with Ukraine, but then went on to “insist” that the country’s president publicly announce investigations into Joseph R. Biden Jr. and his son and other Democrats.

William B. Taylor Jr., the top American diplomat in Ukraine, spoke of his alarm about the conversations during his private testimony last week, saying that he had been briefed about them by Mr. Morrison, the senior director for Europe and Russia for the National Security Council. Mr. Sondland, the ambassador to the European Union, has also given investigators a more limited account of his call with Mr. Trump.

Mr. Morrison’s confirmation of the conversations could be important for House Democrats as they seek to build their impeachment case against Mr. Trump. A publicly available, reconstructed transcript already shows that Mr. Trump pressed President Volodymyr Zelensky of Ukraine during a July 25 phone call to undertake the investigations of Democrats. Investigators are trying to establish whether Mr. Trump used $391 million in security aid and a coveted White House meeting with Mr. Zelensky as leverage in a pressure campaign to secure the inquiries.