It would be the first summit in more than a decade between Korean leaders.
Mr Moon said the Koreas should “make it happen” and encouraged the north to return to negotiations with the US.
The handwritten invitation was delivered by Mr Kim’s influential sister, Kim Yo-jong, at a landmark meeting in the presidential palace in Seoul, before the Winter Olympics.
Ms Kim and the north’s ceremonial head of state Kim Yong-nam made up the most senior delegation from the north to visit the south since the Korean War in the 1950s.
The figures from the two Koreas shared kimchi (Korean pickled cabbage) and soju rice liquor, and spoke for three hours.
Ms Kim invited Mr Moon to visit “at the early date possible”, a spokesman for South Korea’s presidential palace said.
The meeting comes after the US warned against engagement with Pyongyang.
The Trump administration is cautious of Seoul falling for North Korea’s charm offensive during the Winter Olympics, which are taking place amid tension over Pyongyang’s nuclear programme.
Correspondents say the invitation puts Mr Moon in a difficult position as he campaigned on a promise to engage with the North, but his moves in that direction go contrary to the wishes of his ally the US.
Earlier, US Vice-President Mike Pence briefly encountered Kim Yong-nam at the Games but the two tried to avoid directly facing each other, Yonhap news agency reports.
At the opening ceremony for the Games, Mr Pence, Kim Yo-jong and Kim Yong-nam were seated in close proximity to each other.
Mr Pence stayed seated when the athletes of the host nation marched in to the arena alongside those from the north.
Later on Saturday Mr Moon and Mr Kim will attend the first ice hockey match played by athletes from both North and South Korea on Saturday, according to reports.
It was not immediately reported whether Ms Kim would attend the evening match against Switzerland.
The highest profile member of the North Korean delegation to the Games, she is the first immediate member of the North’s ruling family to visit the South since the 1950-1953 Korean war.
Ms Kim, who is said to be very close to her brother, was promoted to the North’s powerful politburo last year.
She is on a US sanctions list over alleged links to human rights abuses in North Korea.
Ms Kim is thought to be about 30 years old, around four years younger than her brother.
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