WASHINGTON—A top White House official for the first time linked President Trump’s decision to hold up aid to Ukraine to his desire for Kyiv to probe a Democratic National Committee server related to the 2016 election, then hours later issued a statement contradicting his own remarks.

The initial comments at a White House briefing by acting chief of staff Mick Mulvaney caused alarm among senior White House aides and the president’s legal team. Mr. Trump has repeatedly denied there was any quid pro quo related to the aid. Late...