People close to Donald Trump have made overtures to media executives in recent months about the possibility of launching a media venture to capitalize on the Republican presidential candidate’s appeal, people familiar with the matter say.

Jared Kushner, Mr. Trump’s son-in-law who is also a close adviser to the GOP nominee, reached out to prominent investment banker Aryeh Bourkoff within the past few months to discuss the prospect of creating such a business after the election, the people said.

Mr....