The challenge is stark. No nation that has openly tested a nuclear device has ever surrendered its arsenal, and North Korea has conducted six underground explosions, each more powerful than the last, and has test-fired missiles that can reach the mainland United States.

But Mr. Trump and Mr. Kim have both already defied conventional wisdom about what is possible. If they meet in June — most likely in Singapore, according to American and South Korean officials — it would be the first direct encounter between the leaders of the two nations, as well as a chance to test the argument that making progress with North Korea in the nuclear standoff requires starting at the top.

The meeting between Mr. Kim and Mr. Moon is the third summit meeting between leaders of the two countries, but the first in which denuclearizing the Korean Peninsula tops the agenda.

“I hope we can have open-minded talks on issues of concern and produce good results, not the kind of results we saw in the past that were not implemented and made us start from scratch again,” Mr. Kim said as the talks began on Friday. For his part, Mr. Moon said he hoped that the two leaders could engage in “broad-minded” discussions and produce “a big gift” for those yearning for peace on the Korean Peninsula.

Mr. Moon hopes to emerge from Friday’s summit meeting with a formal but vague denuclearization commitment from Mr. Kim and perhaps a path to negotiating a peace treaty or a plan to reduce military tensions. Some have suggested a pullback of troops from the Demilitarized Zone between the North and South is possible.

But Mr. Moon has acknowledged that there is a limit to what the two Koreas can agree on without American involvement. “Peace on the Korean Peninsula cannot be achieved by agreements between South and North Korea alone,” he said last month. “It has to have American endorsement.”