WASHINGTON—President Trump wants South Korea to pay significantly more money for American troops stationed in South Korea, a demand that has snarled negotiations over a defense pact as the Seoul government resists, according to people familiar with the talks.

The standoff puts a key American alliance under pressure at a time when the U.S. is pushing for the denuclearization of North Korea, and could weaken U.S. standing with Seoul as it pursues rapprochement with Pyongyang, its neighbor to the north. South Korean President...