It isn't clear if the White House would support such a meeting.
The recent detente, anchored by South Korea's hosting of the Winter Olympic Games that began on Friday, came despite an acceleration in the North's weapons programs previous year and pressure from Seoul's allies in Washington. Moon also met with the sister, Kim Yo Jong, over lunch Saturday.
Moon has already put a summit offer on the table.
Not at all have been cheered by the talks.
After months of silence on whether it would even take part in the Pyeongchang Winter Olympics in the South, which had their opening ceremony Friday, the Games have driven a rapprochement on the peninsula, while the North's athletes, performers, and delegates have dominated the headlines.
Many consider Kim to be the most powerful woman in North Korea.
Past "charm offensives" have been interpreted as North Korea trying to recoup from crippling sanctions on their nuclear program, or trying to drive a wedge between Seoul and its US ally. They were stopped and most were killed near the palace, which was newly rebuilt in 1991.
Kim Jong Il travelled to Beijing to meet then Chinese president Jiang Zemin before the summit. Moon had earlier met Kim Yong Nam during a dinner he hosted for visiting dignitaries.
North Korean player Jong, who had one of the team's eight shots on the night, said she would like to see a unified team continue after the Olympics. They were seated beside the Moons and next to Japanese Prime Minister Shinzo Abe and his wife.
Members of the delegations exchanged greetings during Friday's openg cerermony. "South Koreans are not fools, and neither are the American citizens". In it, Kim Jong Un highlighted several huge intercontinental ballistic missiles, which were successfully flight tested three times past year and could reach deep into the US mainland when perfected.
But the agreement made little headway after the North carried out a series of nuclear and missile tests, and after conservative presidents took office in the South, adopting a harder line on Pyongyang.
Tensions on the Korean border have escalated over the past year as the North pursues its ambition of becoming a fully-fledged nuclear power.
Pyongyang toned down its rhetoric and sent not only athletes to compete in the winter Olympics in South Korea but even dispatched its nominal head of state to the opening ceremony. The fact that the nomination of Ambassador-designate Victor Cha, a former foreign policy advisor to the Bush administration, was withdrawn because of his public opposition to the "bloody nose" strategy has only raised further alarm in South Korea.
Pence also reportedly avoided North Korea's delegation at a VIP reception prior to the ceremony.
The short respite provided by the Olympics will likely end in early March, when the U.S. plans to resume joint military exercises with South Korea.