SEOUL—Google Inc. Executive Chairman Eric Schmidt plans to join an unofficial U.S. delegation on a trip to North Korea this month—the highest-profile American business executive to plan to visit the reclusive country in several years—over objections of the U.S. State Department.

Mr. Schmidt plans to visit Pyongyang this month with Bill Richardson, U.S. ambassador to the United Nations during the Clinton administration and a former governor of New Mexico, and his longtime aide Kun Namkung, according to a person familiar with...