The chief executive of Cambridge Analytica contacted the founder of WikiLeaks to ask him to share Hillary Clinton-related emails at the same time that people familiar with the matter say the British data-analytics firm had begun working for President Donald Trump’s campaign.

Cambridge Analytica CEO Alexander Nix said Thursday he asked the office that handles his speaking engagements to contact WikiLeaks founder Julian Assange in “early June 2016,” after reading a newspaper report that WikiLeaks planned to publish a trove of...