The two North Koreans are on a landmark visit to the South amid a flurry of abrupt reconciliation steps, and both attended the opening ceremony before having a luncheon with Moon at Moon's presidential palace earlier Saturday.
Despite favorable coverage in the South Korean media that largely ignored the horrors inflicted on North Koreans by Kim's regime, many ordinary South Koreans were skeptical of the North's motives.
Such a meeting, if it came about, would mark the first inter-Korea summit since 2007.
The North Koreans might want South Korea and the United States to make concessions if they accede to this basic framework, such as delaying the U.S. "We are making sure our systems are secure and they are secure", said a spokesman for the International Olympic Committee, adding that it's their objective to maintain secure operations.
Pence is viewing the short-track competition, in which eight Americans are participating.
Kim Ryon Hyang of North Korea takes a chair lift to participate in free skiing on the giant slalom course at the Yongpyong Alpine Center at the 2018 Winter Olympics in Pyeongchang, South Korea, Sunday, Feb. 11, 2018.
Ahead of the opening ceremony, South Korea's president, Moon Jae-in, and North Korean leader Kim Jong-un's sister, Kim Yo-jong, exchanged a historic handshake and spoke briefly.
It was the USA that appeared to be the one left in the cold, especially after the sister of the North Korean dictator extended an invitation from her brother for Moon to visit the North. If peace isn't imminent, a summit in Pyongyang between Moon and Kim Jong Un seems preferable to recent months' threats.
Pence has also spent the days leading up to the Olympics highlighting North Korea's human rights abuses and cross-border military provocations - particularly when he met with North Korean defectors and visited the memorial of Cheonan vessel, which is widely believed to have been torpedoed by the North in 2010.
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"The North Koreans should understand that for a summit or any kind of serious talks to occur, Moon needs to be able to take something to Washington - something that addresses denuclearisation", he said.
"It is absolutely necessary for the North and the United States to engage in talks at an early date", he cited Moon as saying.
US Vice President Mike Pence, who was seated in the same box, did not interact with the North Koreans at any point, US officials said.
Moon shook hands with both Kim Yo Jong and Kim Yong Nam at the Olympics opening ceremony on Friday and they cheered as athletes from North and South entered the arena together behind a unification flag showing an undivided Korean peninsula.
A senior White House official described Pence's tact as ignoring the North Koreans, but not avoiding them.
The officials toasted with the Korean liquor "soju" and were served buckwheat and potato pancakes, dried pollock soup, grilled halibut and other dishes.
After sitting at a table, Kim Yo Jong placed a blue document folder in front of her.
Kim Yo Jong, 28, is the first member of the ruling Kim family bearing the bloodline of the sacred Mount Paektu, a centrepiece of the North's idolisation and propaganda campaign, to cross the border into the South since the 1950-53 Korean War.
A rare invitation to Pyongyang for South Korea's president marked Day Two of the North Korean Kim dynasty's southern road tour Saturday, part of an accelerating diplomatic thaw that included some Korean liquor over lunch and the shared joy of watching a "unified" Korea team play hockey at the Olympics.