Officials in Mr. Moon’s government have often said that Mr. Trump’s unconventional approach may offer the best chance for diplomatically resolving the North Korean nuclear crisis, a problem that successive United States administrations have failed to address successfully.

But South Koreans have learned how unpredictable it can be to maintain the alliance with Washington under Mr. Trump, as the American leader swung from threats to rain down “fire and fury” on the North to becoming the first sitting American president to meet the North Korean leader.

But even after Mr. Trump and Mr. Kim met in Singapore and agreed to work toward the “complete denuclearization of the Korean Peninsula” and to build “new” bilateral relations, South Koreans have lived with an abiding fear about the fragility of the diplomatic process.

They express concern that Mr. Trump may change his mind at any time, particularly if he realizes that he cannot deliver the quick denuclearization of North Korea that he seemed to promise to the American people when he declared right after the Singapore summit meeting that there was “no longer a nuclear threat from North Korea.”

South Koreans also worry that Mr. Trump will be distracted by his mounting legal problems at home or that he may not survive for a second term. A diplomatic process that Mr. Moon helped create could be quickly overturned, that line of thinking goes, if a new administration were to take office in Washington with a more skeptical take on diplomatically engaging the North.

When Mr. Moon met with Mr. Kim twice this year, he emphasized how important it was for North Korea to move quickly to reach and carry out a denuclearization deal in return for improved ties with Washington while Mr. Trump was in office, so that it would be hard for future administrations to reverse the course.

But things have not moved as quickly as Mr. Moon had hoped.

Two months after the Singapore summit meeting, talks remain stalemated. Washington wants the North to make bolder steps toward denuclearization, such as revealing a full inventory of its nuclear assets and starting to dismantle some of them.