WASHINGTON—President Trump asked his acting chief of staff to place a hold on $391 million in aid to Ukraine more than a week before a July phone call in which he urged his Ukrainian counterpart to investigate Joe Biden’s son, according to a person familiar with the matter, a revelation that comes as an investigation into the president’s dealings with Kiev is mushrooming on Capitol Hill.

Congress had approved sending military aid to Ukraine to help the country defend against Russian aggression. But Mr. Trump and his advisers,...