An internal White House review of strategy on North Korea includes the possibility of military force or regime change to blunt the country’s nuclear-weapons threat, people familiar with the process said, a prospect that has some U.S. allies in the region on edge.

While President Donald Trump has taken steps to reassure allies that he won’t abandon agreements that have underpinned decades of U.S. policy on Asia, his pledge that Pyongyang would be stopped from ever testing an intercontinental ballistic missile—coupled with the...