Kim's sister Kim Yo Jong, who is visiting South Korea as his special envoy for the Winter Olympics, conveyed her brother's invitation during a meeting at the presidential Blue House, officials from the Blue House said.
Pence left South Korea Saturday night, flying out of Pyeongchang after attending a short-track speedskating event with Moon - who then went to watch ice hockey along with Kim and the North Korean delegation.
He also insisted that Washington was on the same page with its allies, saying "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 ballistic missile program". If Moon accepts, it would be the third round of such talks since their division in 1945.
Asked about criticism that Pence stood and cheered only for Team USA and stayed seated when the North and South Koreans marched in under a unified flag, the official was unapologetic.
And Pence's personal guest at the games will be Fred Warmbier, the father of Otto Warmbier, an American who died past year days after his release from captivity in North Korea.
Mr Moon's spokesman Kim Eui-kyeom said that Mr Kim's sister verbally delivered his offer during a lunch meeting with Mr Moon at Seoul's presidential palace.
USA officials have been urging South Korea to be cautious in its rapprochement with the North.
WEEI suspends Christian Fauria after fake Asian Accent in Don Yee impersonation
After he reads part of the text exchange in the "over the top" accent, his co-hosts played audio of Yee himself. Ex-Patriots tight end Christian Fauria says he reached out to Don Yee to apologize for racist impersonation. .
Several experts here saw the move as a message from the North of its desire to mend ties with South Korea, as Kim Yo-jong's remarks could be taken to directly represent Kim Jong-un's current position on inter-Korean relations, given her ties to the young dictator.
Any summit between the two still-officially warring Koreas would be a coup for Moon, who has been pushing for a diplomatic solution to the standoff over North Korea's nuclear and missile programmes.
At a Friday VIP reception for delegation leaders, Pence arrived late and stayed for just five minutes - and did not interact with the delegation from the North. Speaking to the U.S. and Japanese troops at the Yokota Air Base outside Tokyo, Pence vowed to continue "maximum pressure campaign" on North Korea in a bid to forcing Pyongyang to give up its nuclear missile programme.
These statements, come, as the Olympic Games in South Korea's Pyeongchang got underway.
The two Koreas have been divided since the conflict ended in a ceasefire in 1953, and the democratic South has risen to become the world's 11th-largest economy, while the North has stagnated under the Kim family's rule.
Kim Jong Un dispatched his 30-year-old sister, Kim Yo Jong, as part of the delegation.