He added that he "wanted the honor of meeting with the men and women who fled the tyranny of North Korea".
Kim, president of the Presidium of the Supreme People's Assembly, is leading North Korea's delegation to the Olympics.
President Trump has traded insults with North Korea's leader and has led efforts to slap sanctions on the country for its refusal to yield to worldwide pressure to halt its nuclear weapons program. Jo Yong Sam, department director general of North Korea's foreign ministry, said Thursday the country had never "begged the USA for dialogue" and wasn't about to start now.
"I am very confident, as President Trump is, that President Moon will continue to stand strongly with us in our extreme-pressure campaign", Pence told NBC in an interview, maintaining all options were open to deal with the crisis.
The PyeongChang Olympic Games officially kicked off with the Opening Ceremony on February 9. The North Koreans were present at the reception at the same time as Pence. The combined meetings took almost three hours.
Protesters scuffled briefly with police and criticized South Korean President Moon Jae-in, who is using the Olympics to re-engage with the North and pave the way for talks over the reclusive state's nuclear and missile program.
Officials traveling with Pence said it would be fair to characterize his failure to interact with the North Koreans as a mutual decision, with neither side making any public overtures.
The last summit between leaders of the two Koreas occurred in 2007 when South Korean President Roh Moo Hyun and Kim Jong-il met in Pyongyang, North Korea's capital. "It shows everyone we can all work together".
Although Vice President Mike Pence repeatedly floated the prospect of meeting with North Korean leaders in South Korea, he apparently passed up several opportunities at the Winter Olympics in Pyeongchang.
The spokesman says Moon also called for a quick resumption of dialogue between the United States and North Korea.
"Pence sat in front row and talked to Moon and Abe and their spouses, and the North Koreans sat in the back and didn't talk to anybody, and that image is telling", the officials said.
Seung Yeob Woo, a well-known expert in how to survive a nuclear blast in Seoul, carries an emergency backpack at all times in case of a sudden North Korean attack. "I just don't think you talk geopolitics over speed skating".
Pence stood to welcome the United States athletes as the Korean pop hit Gangnam Style blared around the stadium, sparking the "Horse Dance" in the crowd and among the volunteers.
Some were holding up a ripped-up "Korean Unification" flag, which was used when athletes from South and North Korea made a joint entrance at the opening ceremony of the PyeongChang Olympics on Friday night.