The two leaders emerged from a two-day summit in Pyongyang to sign declarations in front of the world's TV cameras.
If realized, Kim would be the first North Korean leader to visit Seoul since the peninsula was divided into North and South.
The North has carried out several missile test launches from Sohae in the area - its ballistic missile program is banned under UN Security Council resolutions - but has also used many other locations to fire rockets, including the Pyongyang airport.
The North also agreed to take additional steps, such as the permanent shutdown of the Yongbyon nuclear facility, depending on the United States' corresponding measures, he added.
The two Koreas also adopted a separate military accord aimed at preventing armed clashes between the old foes, which are technically still at war because the Korean War ended with a truce, not a peace treaty.
A successful bid by the Koreas would mark the second time South Korea hosted or co-hosted the Summer Games, the first being 1988 in Seoul.
The North's offer to take additional denuclearization steps despite any signs of reward from the United States is apparently aimed at restarting its denuclearization talks with the US.
Pyongyang is said to be demanding early rewards for denuclearization steps it has taken so far, while Washington continues to stress the importance of maximum pressure and sanctions against the North until the communist state fully denuclearizes.
The agreement came in a historic summit between Kim and South Korean President Moon Jae-in held at the state guesthouse Paekhwawon in Pyongyang.
Though North Korea has unilaterally stopped nuclear and missile tests, it did not allowed global inspections for a dismantlement of its only known nuclear test site in May, drawing criticism that its action could not be verified and could be easily reversed.
North Korea has agreed to "permanently" shut one of its main missile facilities in the presence of foreign experts, South Korea's President Moon Jae-in announced at a joint press conference with Kim Jong-un on Wednesday.
The institution was sanctioned by the U.N. Security Council past year as part of global efforts to rein in Pyongyang's nuclear and missile programmes by drying up its revenue sources.
"It's hard to say an agreement has been reached, and we are expecting further dialogue", Mr Moon's press secretary Yoon Young-chan told a news briefing on Wednesday. The facilities were a key test center for its intercontinental ballistic missiles created to reach the United States.
The outcome will also be a litmus test for another meeting Mr Kim has recently proposed to Mr Trump.
North Korea says Kim's grandfather and father were born at Mount Baektu, a centerpiece of the North's idolization and propaganda campaign to highlight the ruling family's sacred bloodline.
The premiere of the latest version of the show, called "The Glorious Country", earlier this month featured video footage of Moon and Kim together at their first summit in Panmunjom - prompting the unusual sight of tens of thousands of North Koreans applauding images of the South's president.
Ms Nauert said the United States would "continue to consult very closely and carefully and regularly with the South Koreans" during their meetings in North Korea.