Moon showed himself willing to visit Pyongyang, and said both Koreas should "accomplish this by creating the right conditions".
The delegation includes the North Korean ceremonial head of state Kim Yong Nam, who has attended different global games before, and Kim Yo-jong, Kim Jong-un's younger sister.
Some 800 anti-North Korean protesters marched and staged street performances in front of the Deoksu Palace in central Seoul, condemning the North Korean leader and President Moon Jae-in, who they regard as too soft on the North.
This is the first time a member of the Kim family visits the south since the peninsula was divided in two.
North Korea has emerged as the early favorite to grab one of the Winter Olympics' most important medals: the diplomatic gold.
Mr Moon has previously said he will need reassurance from the North that it would help resolve the nuclear crisis.
He also urged North Korea to engage in talks with the United States. A Blue House official initially said the president had accepted Kim's offer, but that was later denied by chief spokesman Yoon Young-chan.
And at the Olympics itself, the North and South are competing as a unified "All Korean" team - marching under one flag at the Opening Ceremony and playing together at the ice hockey tournament.
Yong-nam is the President of the Presidium of the Supreme People's Assembly of North Korea.
On Friday, Pence said the USA will be working to come up with a peaceful solution, but added that it will oppose talks between South and North Korea that don't involve the North dismantling its nuclear program.