0951 GMT February 16, 2020
Moon swept to power last year on a policy of engaging more with the North and has pushed for a diplomatic solution to the standoff over North Korea's nuclear and missile program, Reuters reported.
The recent detente, anchored by South Korea's hosting of the Winter Olympic Games that began on Friday, came despite an acceleration in the North's weapons programs last year and pressure from Seoul's allies in Washington.
The personal invitation from Kim was delivered verbally by his younger sister, Kim Yo Jong, during talks and a lunch Moon hosted at the presidential Blue House in Seoul.
Kim Jong-un wanted to meet Moon "in the near future" and would like for him to visit North Korea "at his earliest convenience", his sister told Moon, who had said "let's create the environment for that to be able to happen," Blue House spokesman Kim Eui-kyeom told a news briefing.
A Blue House official said Moon "practically accepted" the invitation.
"We would like to see you at an early date in Pyongyang", Kim Yo Jong told Moon during the lunch, and also delivered her brother's personal letter that expressed his "desire to improve inter-Korean relations," the Blue House said.
The prospect of two-way talks between the Koreas, however, may not be welcomed by the United States.
Washington has pursued a strategy of exerting maximum pressure on Pyongyang through tough sanctions and harsh rhetoric, demanding it give up its pursuit of nuclear weapons first for any dialogue to occur.
Moon asked the North Korean delegation during Saturday's meeting to more actively seek dialogue with the United States, saying that "early resumption of dialogue (between the two) is absolutely necessary for developments in the inter-Korean relations as well," the South said.
It said the two sides held "a comprehensive discussion ... on the inter-Korean relations and various issues on the Korean peninsula in an amicable atmosphere," but did not say whether the North's weapons program was mentioned.
A visit by Moon to the North would enable the first summit between leaders from the two Koreas since 2007, and would mark only the third inter-Korean summit to take place.
Pyongyang conducted its largest nuclear test last year and in November tested its most advanced intercontinental ballistic missile that experts said has the range to reach anywhere in the United States.
US President Donald Trump and the North Korean leadership traded insults and threats of nuclear war as tensions rose, with Trump repeatedly dismissing the prospect or value of talks with North Korea.
US Vice President Mike Pence, who had attended the opening ceremony, said the United States, South Korea and Japan were in complete agreement on isolating Pyongyang over its nuclear weapons program.
"There is no daylight between the United States, the Republic of Korea and Japan on the need to continue to isolate North Korea economically and diplomatically until they abandon their nuclear and ballistic missile program," Pence told reporters on his flight back to the United States.