Early this year, President Donald Trump’s son-in-law and aide, Jared Kushner, met a top executive at CNN parent Time Warner Inc. and raised concerns about the network’s coverage of the presidential election.

Mr. Kushner told the executive, Gary Ginsberg, that CNN should fire 20% of its staff because they were so wrong in their analysis of the election and how it would turn out, people familiar with the matter say.

A...