Such a summit, if it came about, would mark the first time that leaders of the two Koreas have met since 2007.
Pence - who sat just feet away from Kim Yo Jong at the Olympic opening ceremony, with neither apparently speaking to the other - has also been critical of the North's participation in the games, seeing it as an attempt to create a division between the United States and the South.
Moon and the two North Korean delegates cheerfully clapped and waved as the athletes from the two Koreas jointly marched during the ceremony holding a blue-and-white flag symbolizing a unified Korean Peninsula.
US Vice President Mike Pence, who has left South Korea for Washington, said the United States, South Korea and Japan were in complete agreement on isolating North Korea over its nuclear weapons programme.
It was the US 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.
"I am very confident, as President Trump is, that President Moon will continue to stand strongly with us in our extreme-pressure campaign", Pence told NBC in an interview, maintaining all options were open to deal with the crisis.
He said the secretary-general was also in "the same general area" as a number of other officials from Pyongyang including North Korean leader Kim Jong Un's younger sister, Kim Yo Jong, "but they were never in any personal contact with each other".
North Korean supporters cheer during the preliminary round of the women's hockey game between Switzerland and the united Korean team.
Loud applause greeted the appearance of Moon, Kim Yo Jong and her delegation at the concert, attended by more than a thousand people.
South Korea accommodated both the North Korean government officials and members of the art troupe at the Walkerhill hotel in Seoul. If peace isn't imminent, a summit in Pyongyang between Moon and Kim Jong Un seems preferable to recent months' threats.
The North Korean Leader started the New Year by letting it be known that he is in possession of a "nuclear button" on his desk that he's not afraid to use in the event that the us provokes him.
Choi Bok-mu, a fitness club manager at the Olympics athletes village in Gangneung, said he had not experienced any problems communicating with North Korean athletes, despite them speaking in a markedly different accent.
Pence has repeatedly said he would deliver a tough message to the North in any meeting.
"I would love the team to get the Nobel Peace Prize", Ruggiero, a member of the IOC's executive board said a day after the unified Korean team competed at the Pyeongchang Olympics.
"I never thought I would visit (the South) so suddenly and believed much would be odd and different but I saw many things that were similar or the same", said Kim Yo Jong in a toast during Sunday's dinner, adding she hoped to meet the "friendly faces" before her later in Pyongyang.
Over the last few days the world has witnessed unprecedented acts of cooperation, symbolic or otherwise, between South Korea and its nuclear-armed neighbor to the North.
Accepting North Korea's demand to transport more than 100 members of the art troupe by sea, South Korea treated the Mangyongbong-92 ferry as an exemption to the maritime sanctions it imposed on the North, a controversial move amid concerns that the North is trying to use the Olympics to poke holes in worldwide sanctions. "It is hard to imagine making Kim's face into a mask in North Korea, where losing a Kim Il-sung badge was a sin that could land you in a concentration camp", Cha Du-hyeogn, a visiting research fellow at the Asan Institute for Policy Studies, wrote on his Facebook page.
"We've been told not to talk about nukes or missiles before we came here", Choi said.
"We are determined to make sure that even in the midst of the powerful background & idealism of the Olympics, the World is reminded of the truth about North Korea", Pence tweeted on Friday.
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