The delegation, which includes Kim's sister, Kim Yo-jong, is in South Korea for the Winter Olympics.
In stark contrast, two kinds of kimchi - the fermented cabbage that features in every Korean meal - were on the menu for lunch Saturday, one mild Northern style version and a spicier Southern recipe, a Blue House official told AFP, along with soju, the traditional Korean rice liquor.
Prime Minister Lee Nak-yon (R) shakes hands with Kim Yo-jong, sister of North Korean leader Kim Jong-un, during a lunch meeting in Seoul on February 11, 2018.
It would be the first summit between the leaders of the two Koreas in more than a decade.
Moon, who returned to the Games venue, joined Pence in the arena and sat next to him, turned around to greet Warmbier, according to a White House pool report.
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To pave the way for the North's participation at Pyeongchang, South Korea had delayed the annual exercises with US forces, which usually take place between February and March, until after the Olympics.
The North Korean diplomatic initiative comes amid growing worldwide pressure, led by the United States, aimed at imposing maximum economic and diplomatic pressure on the regime.
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.
Lee also said that inter-Korean talks should continue after the Olympics. The United States fought with South Korea and maintains tens of thousands of troops and an "ironclad" agreement to protect its ally. Pence carried a similar message during his trip to Japan and South Korea.
"My message in Japan, my message in South Korea, my message in meeting with leaders of both of those countries is that we are going to continue to stand together along with our other allies and partners to continue to intensify the economic and diplomatic pressure on North Korea until they permanently abandon their nuclear and ballistic-missile program", Pence said.
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.
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Some North Korean experts believe tough United Nations sanctions that are cutting off most of the isolated North's sources of revenue have added pressure on Pyongyang to engage further with Seoul.
Pyongyang conducted its largest nuclear test previous year and in November tested its most advanced intercontinental ballistic missile that experts said has the range to reach anywhere in the United States.
A spokesperson said: "The vice president is grateful that President Moon reaffirmed his strong commitment to the global maximum pressure campaign and for his support for continued sanctions".
Pence said the United States and South Korea were closely aligned in their approach to dealing with North Korea.
The two Koreas have a rocky and sometimes violent history at the Blue House.
However, no member of North Korea's ruling family has ever before visited the Blue House, which is near where North Korea founder Kim Il Sung - Kim Yo Jong's grandfather - sent dozens of commandos to assassinate former South Korean military strongman Park Chung-hee in 1968.
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Pence and the North Korean delegation, who both attended the Games opening ceremony, had no contact with each other. They may also see a performance by a visiting North Korean art troupe in Seoul on Sunday before flying back to Pyongyang.