The Winter Games are being staged only 80km (50 miles) from the border with North Korea, which is technically still at war with the South since their 1950-1953 war ended in a truce rather than a peace treaty.
The delegation gave Moon a letter from North Korean leader Kim Jong Un inviting him to visit the North, a day after Prime Minister Shinzo Abe said he and Moon had reaffirmed their shared stance of maximizing pressure on Pyongyang until it scraps its nuclear weapons program.
Moon has said he would be willing to visit Pyongyang to talk directly with Kim Jong Un about cooling off the country's nuclear program.
"I do the torch relay now for the seventh time, but every time it's like a first time".
The Winter Olympics have grabbed global spotlight not just because of the sport but also because of the diplomatic affinity between North Korea and South Korea shown during the games.
South Korea's president has received an invitation to visit the North - an offer that came from the younger sister of the North's dictator.
Moon reportedly responded saying "an early resumption of dialogue between the United States and the North is needed also for the development of the South-North Korean relationship".
"I never thought I would visit (the South) so suddenly and believed much would be odd and different but I saw many things that were similar or the same", said Kim Yo Jong in a toast during Sunday's dinner, adding she hoped to meet the "friendly faces" before her later in Pyongyang.
Blue House spokesman Kim Eui-kyeom cited Moon as replying: "Let's create the environment for that to be able to happen".
The Games, whose opening ceremony will takes place at 8 p.m., on Friday, are kicking off amid increased political attention surrounding the participation of North Korean athletes and the potential of improved ties on the Korean peninsula.
"What can not be overlooked is that Moon did not directly demand North Korea to abandon its nuclear development".
The North Korean delegation was invited to lunch with South Korean Prime Minister Lee Nak-yeon at the five-star Walkerhill Hotel in eastern Seoul on Sunday, an official at the prime minister's office told Reuters.
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