NASHVILLE, Tenn.—The Securities and Exchange Commission wants to make it easier for individuals to invest in private companies, including some of the world’s hottest startups, the agency’s chairman said in an interview.

SEC Chairman Jay Clayton, a Trump appointee wrestling with how to boost flagging interest in public markets, said the commission also wants to take steps to give more individual investors a shot at companies that have been out of their reach because they haven’t gone public.

Companies...