SEOUL—Nearly three months after North Korean leader Kim Jong Un shook hands with President Trump and agreed to pursue denuclearization and a reset in relations, the two sides have reached an impasse over who should make the next concession in talks aimed at implementing their accord.

The standoff threatens to derail a delicate detente with Pyongyang, following weeks of negotiations in which both sides say they have given ground but which so far haven’t produced a clear agreement on disarmament steps or moves toward a peace...