While the North Korean dictator, who has yet to meet a foreign leader, has invited the South Korean president for a rare summit, Kim has given no sign of being ready to talk to the U.S.
The move to bring the two nations together for the ice hockey tournament has not been universally popular in the Winter Olympic host nation, while it almost collapsed on the eve of the Games when North Korea threatened to withdraw following comments made by United States Vice-President Mike Pence.
The handwritten invitation was delivered by Mr Kim's influential sister, Kim Yo-jong, at a landmark meeting in the presidential palace in Seoul, before the Winter Olympics.
The event also witnessed South Korean president Moon Jae-in sharing the VIP box with North Korean representatives, which included Kim Yo-jong, sister of the supreme leader who has been at loggerheads with Trump after the two exchanged nuclear threats.
"For the sake of advancing relations between the two Koreas, it is imperative for the North and the United States to talk soon", Moon was quoted as saying by his spokesman.
Pence added that he and Moon assured each other "that we will continue to stand strong and work in a coordinated way to bring maximum economic and diplomatic pressure to bear on North Korea". Jo Yong Sam, department director general of North Korea's foreign ministry, said Thursday the country had never "begged the U.S. for dialogue" and wasn't about to start now.
Moon asked the North Korean delegation during Saturday's meeting to engage in dialogue with Washington "at an early date", the Blue House said. Kim Yo Jong is 28 years old, according to the USA government.
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A visit by Moon to the North would be only the third inter-Korean summit to take place.
Pence shook hands with other leaders, including close ally Japanese Prime Minister Shinzo Abe, but not Kim Yong-nam, according to a Blue House pool official.
The US has imposed sanctions on North Korea over the latter's nuclear weapons development.
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.
Those are the flags of united Korea that athletes from North Korea and South Korea marched under during Friday's Opening Ceremony. The previous meetings came after rounds of global talks aimed at eliminating the North's nuclear program, which eventually failed. But Kim Jong Un has recently proclaimed that his country is now a "nuclear state in its own right". At one point the North Korean cheerleaders unsuccessfully tried to get the crowd of about 3600 to do the wave. Days later, TV news shows in South Korea are still repeatedly airing footage of Kim walking with President Moon Jae-in, often with a placid smile on her face.