Trump donor Rebekah Mercer in August 2016 asked the chief executive of a data-analytics firm working for Donald Trump’s presidential campaign whether the company could better organize the Hillary Clinton-related emails being released by WikiLeaks, according to a person familiar with their email exchange.

The previously undisclosed details from the exchange between Ms. Mercer and Cambridge Analytica CEO Alexander Nix show how an influential Trump supporter was looking to leverage the hacked Clinton-related messages to boost...