In the summit with Trump, Moon stressed the need for the United States to take corresponding measures for the denuclearization steps taken by the North.

Since the historic US-North Korea summit held in Singapore in June, the communist state has dismantled its only nuclear test site in Punggye-ri.

In his third bilateral summit with Moon, the North Korean leader offered to take additional steps, including the permanent dismantlement of the country's missile engine test facility and missile launch pads.

"The leaders agreed to seek ways to encourage North Korea's denuclearization process by showing the bright future it may enjoy once it reaches complete denuclearization," the spokesman of Moon's presidential office Cheong Wa Dae said of the Moon-Trump summit earlier.

Trump has also agreed to meet the North Korean leader for their second bilateral summit in the "not too distant future."

Moon laid out the details of his third inter-Korean summit with Kim in an address to the United Nations this week, urging the international community to help his country's efforts to establish lasting peace on the Korean Peninsula and in the Northeast Asian region.

"North Korea has come out of its long isolation on its own to stand before the world again. Now, it is time for the international community to respond to North Korea's new choice and efforts," he said in his address to the world body.

Moon also stressed the need to formally end the 1950-53 Korean War as a way of providing some assurances to the North.

The two Koreas technically remain at war as the Korean War ended with an armistice, not a peace treaty.

President Moon invited the North Korean leader to visit Seoul in the near future, and Kim accepted the invitation.

"The declaration of an end to the war that the South and the North are seeking is a process we must undergo to move toward a peace regime," Moon said Tuesday while meeting with a group of US opinion leaders.

"It is also needed to accelerate North Korea's denuclearization steps." (Yonhap)