Mr. Moon and Mr. Kim have both said they want to use an opening created by the North’s participation in the Winter Olympics in South Korea last month to improve inter-Korean ties.

But Washington and North Korea remain far apart over the terms under which they would start a dialogue, a gap that South Korea seeks to narrow. After returning home from Pyongyang, the South Korean envoys will fly to Washington for discussions with the Trump administration.

The Trump administration says it is determined not to repeat what it calls the mistakes of its predecessors, who tried both dialogue and sanctions but failed to stop the North’s nuclear program. The administration now says it will enter negotiations with North Korea only after it commits to discussing denuclearization.

American officials fear that North Korea is more interested in weakening sanctions that have begun biting the impoverished country than engaging in serious negotiations. Even if talks begin, they say, Washington will not stop its campaign of “maximum” pressure and sanctions until the North dismantles its nuclear weapons program.

North Korea rejects any preconditions for talks, saying Washington must treat it like an “equal” nuclear power. It also insists that any talks with Washington would have to deal not only with its nuclear program but also with “hostile” American policies, like the United States’ annual joint military exercises with the South, which the North says forced it to seek a nuclear deterrent in the first place.

Mr. Moon spent most of the last year helplessly watching the Korean Peninsula edge toward a possible war, as the North test-launched intercontinental ballistic missiles and conducted its most powerful nuclear test, and Mr. Trump threatened to rain down “fire and fury” on North Korea.

Mr. Moon saw an opening when Mr. Kim agreed to send North Korean envoys, as well as athletes and cheerleaders, to the Olympics. He has since assumed the role of a matchmaker in persuading Washington and Pyongyang to soften their stances enough to make dialogue possible.