Some of President Donald Trump’s lawyers earlier this summer concluded that Jared Kushner should step down as senior White House adviser because of possible legal complications related to a probe of Russia’s involvement in the 2016 presidential election and aired concerns about him to the president, people familiar with the matter said.

Among their concerns was that Mr. Kushner was the adviser closest to the president who had the most dealings with Russian officials and businesspeople during the campaign and transition, some...