United States officials have remained suspicious over Kim's intentions as the North cosies up to the South over the Olympics.
Kim Jong Un wanted to meet Moon "at an early date" and Moon had said "let's create conditions to make it happen", Blue House spokesman Kim Eui-kyeom told a news briefing.
Last year, North Korea conducted dozens of missile launches and its sixth and largest nuclear test in defiance of U.N. Security Council resolutions as it pursues its goal of developing a nuclear-armed missile capable of reaching the United States.
From the South Korean side, the unification minister and Vice Unification Minister Chun Hae-sung participated in the banquet along with Lee Hee-beom, the president of the PyeongChang 2018 Winter Olympic Games organizing committee, and Choi Moon-soon, governor of the Gangwon province where the host city is located.
"We would like to see you at an early date in Pyongyang", Kim Yo Jong told Moon during the lunch, according to another Blue House official.
The South Korean President also stressed the need for the Communist North to resume its dialogue with the US.
The figures from the two Koreas shared kimchi (Korean pickled cabbage) and soju rice liquor, and spoke for three hours.
Pence, who is leading the U.S. delegation to the Olympics, briefly encountered Kim Yong-nam at the Games but the two tried to avoid directly facing each other. "So the maximum pressure campaign is going to continue and intensify".
The two Koreas are still technically at war after the 1950-53 conflict on the Korean peninsula ended in a ceasefire and not a truce. Kim Yong Nam and South Korean President Moon Jae-in were expected to watch the game together in Gangneung.
Kim Yo Jong arrived in South Korea on Friday with Kim Yong Nam, the North's nominal head of state, for the opening ceremony in the alpine town of Pyeongchang.Pence also attended the Games opening ceremony but had no contact with the North Korean delegation.
An inter-Korean summit would be the third of its kind, after Kim's father and predecessor Kim Jong Il met the South's Kim Dae-jung and Roh Moo-hyun in 2000 and 2007 respectively, both of them in Pyongyang.
North Korea defends its weapons programmes as essential to counter USA aggression, saying regular war drills between the United States and the South are preparations for invasion.
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