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.
Kim Yo Jong shook hands with Moon at the beginning of the ceremony.
After a chaotic year of nuclear war threats and missile tests from the North, it was a striking visual moment.
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McDonald was previously a member of Fianna Fáil, another Irish republican party, although quit to join Sinn Fein in 1998. She became the party's deputy in 2009 and in 2011 won a seat in the Irish parliament, representing Dublin Central.
The country has had a history of taking provocative acts and agreeing to talks that lead nowhere.
"President Moon responded by saying that the two sides should work on establishing the right conditions to realize the meeting", the spokesperson said.
Pence and the North Koreans have been an awkward presence at the Games to say the least.
North Korea has spent years developing its military, saying it needs to protect itself from United States aggression.
The vice president also met with North Korean defectors while in Pyeongtaek.
North and South Korea are technically still at war after their 1950-53 conflict ended in a truce rather than a peace treaty.
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Ahead of the reception, hosted by South Korean President Moon Jae-in, South Korean media said Pence was expected to be seated opposite Kim Yong Nam, North Korea's nominal head of state, at the 12-seat head table.
Although Kim Yo Jong ranks diplomatically lower than Kim Yong Nam, as Kim Jong Un's sister she is considered more important in North Korea.
While a summit in Pyongyang would signal warming ties on the peninsula, it also risks driving a wedge in the alliance between the US and South Korea. Oh, and a fake Donald Trump and Kim Jong Un, who were promptly escorted out of the stadium.
"From a regular people's point of view, the athletes are nowhere near the general people", Kim Danbi, 28, who defected from North Korea in 2011, said.
"It helps that she also appears dynamic and photogenic", he said.
Kim Yo Jong, 30, was elected to the politburo previous year, but remains little known outside North Korea and has not appeared much in public until now.
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A320neo operators with Pratt & Whitney engines have been plagued by reliability issues since they began flying passengers in 2016. Neither the aircraft manufacturer nor the agency disclosed what airlines are operating aircraft with affected engines.
"She is a bit of a mystery figure", Madden said.
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