The invitation, delivered by Kim's visiting sister Kim Yo Jong, said Kim was willing to meet the South's leader "at the earliest date possible", said a spokesman for the presidential Blue House.
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.
But it could threaten to sow division between Moon, who has long argued for engagement with the nuclear-armed North to bring it to the negotiating table, and US President Donald Trump, who last year traded personal insults and threats of war with Kim.
Washington insists that Pyongyang -- which is under multiple sets of UN Security Council sanctions -- must show a willingness to give up its weapons before any negotiations can happen.
After months of silence on whether it would even take part in the Pyeongchang Winter Olympics in the South, which had their opening ceremony Friday, the Games have driven a rapprochement on the peninsula, while the North's athletes, performers and delegates have dominated the headlines.
Moon met Kim Yo Jong and the North's ceremonial head of state, the elderly Kim Yong Nam -- technically the highest-level Northern official ever to go to the South -- for talks and lunch at the Blue House on Saturday in a landmark meeting.
"Special envoy Kim Yo Jong delivered a personal letter" from her brother stating his desire to "improve inter-Korean relations", said Moon's spokesman Kim Eui-kyeom, and verbally conveyed his offer to Moon "to visit the North at his most convenient time".
Moon did not immediately accept the offer, calling instead for efforts to "create the right conditions to realise" such a visit and urging Pyongyang to actively seek dialogue with the US, he added.
"It is absolutely necessary for the North and the United States to engage in talks at an early date," he cited Moon as saying.
Tensions between the two soared last year as Pyongyang launched intercontinental ballistic missiles capable of reaching the US mainland and carried out by far its most powerful nuclear test to date.
Analysts believe the diplomatic drive by the North -- which put its ICBMs on show at a military parade in Pyongyang on Thursday -- is seeking to weaken the measures against it, and could be trying to loosen the alliance between Seoul and Washington.
Kim Yo Jong has rapidly risen up the ladder since her brother inherited power from their father, and is now among his closest confidantes.
Her visit makes her the first member of the dynasty to set foot in the South since the end of the Korean War in 1953.
'Blind eye'
Moon shook hands with both Kim Yo Jong and Kim Yong Nam at the Olympics opening ceremony and they cheered as athletes from North and South entered the arena together behind a unification flag showing an undivided Korean peninsula.
US Vice President Mike Pence, who was seated in the same box, did not interact with the North Koreans at any point, US officials said.
He also did not shake hands with Kim Yong Nam while making a brief appearance at a leaders' reception ahead of the ceremony -- although Japanese Prime Minister Shinzo Abe, whose country is also regularly threatened by Pyongyang, did so, while exchanging pleasantries with the North Korean.
"The US will not allow the propaganda charade by the North Korean regime to go unchallenged on the world stage," Pence tweeted on Saturday. "The world can NOT turn a blind eye to the oppression & threats of the Kim regime."
Pence has repeatedly said he would deliver a tough message to the North in any meeting.
"At the outset of any new dialogue or negotiations," he said Friday, Pyongyang had to "put denuclearisation on the table and take concrete steps with the world community to dismantle, permanently and irreversibly, their nuclear and ballistic missile programmes.
"Denuclearisation has to be the starting point of any change," he said.
Kimchi and soju
In stark contrast, two kinds of kimchi -- the fermented cabbage that features in every Korean meal -- were on the menu for lunch Saturday, one mild Northern style version and a spicier Southern recipe, a Blue House official told AFP, along with soju, the traditional Korean rice liquor.
The smiles and handshakes at the meeting were friendlier than some of the North's past history with the complex -- in 1968 it sent commandos to attack it to try to assassinate the South's then leader.
The North's official media have reflected the positive tone, with the ruling party's mouthpiece Rodong Sinmun carrying seven pictures of the delegation's departure from Pyongyang and arrival in Incheon on its front page Saturday.
On page two it printed seven more of the opening ceremony and its representatives' meetings and handshakes with Moon, whom it described as president.
It is rare for the North's official media to refer to the South's leader as president, usually describing them as chief executive or similar, and even more unusual for a picture of them to be shown.
The paper lambasted Pence for his "anti-DPRK lunatic spasm which does not fit the spirit of the Olympics".
"We don't do such treacherous and dirty things exploiting sports festivals for political purposes like the US does," it insisted.
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