Any meeting would represent a diplomatic coup for Moon, who swept to power previous year on a policy of engaging more with the reclusive North.
Pence was reported to have said he reached the new understanding with South Korean President Moon Jae-in, who has been pushing for diplomatic solution to the standoff over North Korea's nuclear and missile programs, in two substantive conversations during his visit to South Korea.
Kim Jong Un wanted to meet Moon "at an early date" and Moon had said "let's create conditions to make it happen", Blue House spokesman Kim Eui-kyeom told a news briefing.
Moon did not immediately accept the invitation.
Moon met Kim Yo Jong and the North's ceremonial head of state Kim Yong Nam for talks and lunch at the Blue House today.
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.
The comments represent a departure from the Trump administration's previous stance that it would not engage in any dialogue with North Korea unless Kim Jong Un's regime first agreed to the goal of denuclearization.
Saturday's meeting was the first time a South Korean president hosted North Korean officials for a luncheon at the Blue House since November 2007, when late liberal President Roh Moo-hyun, the political mentor of Moon, invited then-North Korean premier Kim Yong Il following a meeting with government officials in Seoul.
The meeting came amid brisk diplomacy among the members of six-party denuclearisation talks - the United States, China, Japan, Russia and the two Koreas.
"I think it is different from the last 20 years", Pence said, according to the Post.
"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 programme", Pence told reporters during a flight to the United States following his visit to South Korea, where the Olympic games are being held.
The two Koreas are still technically at war after the 1950-53 conflict on the Korean peninsula ended in a ceasefire and not a truce.
The North Koreans went through a busy schedule in South Korea as the world watched their every move.
North Korea has spent years developing its military, saying it needs to protect itself from USA aggression.
After months of silence on whether it would even take part in the Pyeongchang Winter Olympics in the South, the Games have driven a rapprochement on the peninsula, while the North's athletes, performers and delegates have dominated the headlines.
The North has sent almost 500 people to the Pyeongchang Games, including officials, athletes, artists and also a 230-member state-trained cheering group after the Koreas agreed to a series of conciliatory gestures for the games.
South Korean television showed a smiling Moon entering a reception room and shaking hands with the North Koreans, who also included Choe Hwi, chairman of the country's National Sports Guidance Committee, and Ri Son Gwon, chairman of the North's agency that deals with inter-Korean affairs.
Meanwhile, South Korea said it will seek ways to continue engaging North Korea, including trying to arrange more reunions for families divided by the Korean War and aim to lower military tensions. "And he is encouraging both sides to come together".
USA's Mirai Nagasu lands historic triple axel in PyeongChang
Japanese skaters Midori Ito and Mao Asada are the only other women to successfully complete the triple axel at the Olympics. The husband-and-wife pairs team of Alexa Scimeca-Knierim and Chris Knierim placed fourth in their free skate on Sunday.