A visit by Moon to the North would enable the first summit between leaders from the two Koreas since 2007.
Camera flashes; mobs of reporters: Through it all, Kim has maintained a placid smile - and despite Pence's stated intention to reinforce North Korea's isolation during his Olympic visit, the sister of the repressive regime's leader seems to have spent at least as much time with Moon as Pence did.
On Saturday morning, after the opening ceremony, Pence tweeted his agreement with former Ambassador to the United Nations John Bolton, who had said on Fox News that the positioning of North Korea near South Korea at the opening ceremony was a "propaganda charade" by North Korea.
It's by far the most sustained public presence for Kim, who had been something of a mystery to many South Koreans, despite her known role in North Korea's politburo.
A White House official described the South Korean president as hopeful that there would be "some talks and socializing" between Pence and the North Koreans over the course of the weekend.
The North showed off long-range missiles, including the Hwasong-15 intercontinental ballistic missile that may be capable of reaching the USA mainland, in a message that it would not abandon its nuclear and missile programs.
North and South Korea are technically still at war after their 1950-53 conflict ended in a truce rather than a peace treaty. "It's not unfamiliar", Kim said when asked by one of the South Koreans at the dinner how she feels on the other side of the border.
After the opening ceremony, the North Korean delegates moved to Seoul and spent the night at the Walkerhill hotel, a riverside facility named after late U.S. Army commander Walton Walker, who's considered a Korean War hero in the South.
"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".
"The inter-Korean relations should be mended by the parties concerned at any cost", Moon was quoted by the agency as adding.
Lee expressed his hopes to the delegation that the "right conditions" would be created for the two heads of state to meet, the official said later.
Kim Yo Jong is the first member of her family to set foot in this country since the north and south were divided and her grandfather, Kim Il Sung, founded North Korea.
However, South Korea's main opposition party warned that any talks between the two Koreas where the scrapping of North Korea's nuclear program was not a precondition would only "benefit the enemy".
For Pence's supporters, "I think the hard-line wing of the United States thinks he did a fine job", said David C. Kang, director of the Korean Studies Institute at the University of Southern California.
Japanese editorials sounded a similar warning, saying dialogue would be meaningless unless it led to denuclearisation of the Korean peninsula.
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 United States, he added.
Throw in there the accusations that South Korea has had to arrange huge payouts for past meetings, and that these earlier encounters, while producing indelible images, have done little to slow North Korea's pursuit of nuclear weapons.