The invitation was verbally delivered by Kim's sister, Kim Yo Jong, during a Saturday meeting at Moon's presidential compound in Seoul a day after the opening ceremony for the Winter Olympics in South Korea.
Saturday's lunch at the Blue House, as Moon's presidential compound is known in Korea, came shortly after Moon and Kim Jong Un's sister enthusiastically cheered on a unified Korean team as they walked around a stadium in Pyeongchang.
But away from the spotlight, these two unwanted guests, men dressed as North Korea supreme leader and the United States president, were escorted away from the premises.
Song Senung Hwan, executive creative director for the opening ceremony, said he had been able to make this most of his budget which was KRW60 billion (€45million).
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.
The last summit was held in October 2007 between Roh Moo-hyun and Kim Jong Il, the father of the current North Korean leader.
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Moon recalled his past visit to the North's Diamond Mountain resort, where he and his mother met his North Korean aunt during a temporary reunion of families separated by the 1950-53 Korean War. "And so when you put it that way they're not really participating in the Olympics".
The Winter Olympics has thrust Kim Yo-jong into the spotlight. They (the North Koreans) just want to learn and get better in hockey.
He was jailed and was not seen for 18 months until he was suddenly and dramatically given what North Korean officials called humanitarian release because of his rapidly declining health.
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.
Moon has been pressing the North to engage with the USA to de-escalate tensions between the countries, which have been fueled by a string of nuclear tests that were met with strong rhetoric from President Donald Trump. A total of about 500 North Koreans, including 22 athletes and 140 musicians, arrived in South Korea this week for the Olympics.
In a statement, Moon said he planned to set the conditions to allow the talks to happen.
Later in the ceremony, when the two Korean teams marched into the stadium together under a flag showing a unified Korean Peninsula, Pence, his wife Karen Pence and conservative Japanese prime minister Shinzo Abe remained seated, even as everyone else in the box, including Olympic officials, rose to applaud.