South Korean President Moon Jae-in speaks during a news conference at the Presidential Blue House in Seoul, South Korea, May 27, 2018. -North Korea summit, and the strongest sign yet that the two Koreas' leaders are trying to keep the meeting on track.
Mr Trump had rattled the region on Thursday by cancelling his meeting with Mr Kim, citing "open hostility" from the Pyongyang regime.
The Washington Post previously reported on the team entering into North Korea.
Rekindling hopes, Trump said it was even possible the meeting could take place on the originally planned June 12 date.
Saturday's Korean summit came hours after South Korea expressed relief over revived talks for a Trump-Kim meeting.
At their first meeting on April 27, Kim and Moon first announced "complete denuclearisation of the Korean Peninsula" and permanent peace, which Seoul has tried to sell as a meaningful breakthrough to set up the summit with Trump.
President Donald Trump is suggesting the potentially historic North Korean summit he had suddenly called off might be getting back on track. "It's moving along very nicely". So we're looking at June 12th in Singapore.
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"We believe that, as a party to the [Korean Peninsula's] nuclear issue, the DPRK and U.S. leaders' meeting will play a key role in promoting the denuclearisation of the peninsula", he added. Moon said on Sunday that the North's disarmament could be still be a hard process even if Pyongyang, Washington and Seoul don't differ over what "complete denuclearisation" of the peninsula means. Japan, the United States, and South Korea will continue to put pressure on North Korea to change its policy.
Kim and Trump's initial decision to meet followed months of war threats and insults between the leaders over the North's nuclear program.
"What Kim is unclear about is that he has concerns about whether his country can surely trust the United States over its promise to end hostile relations (with North Korea) and provide a security guarantee if they do denuclearization", Moon cautioned.
The weekend talks were the latest twist in a week of diplomatic ups and downs over the prospects for an unprecedented U.S. "I urge North Korea and the U.S.to confirm each other's will by sharing their problems with each other and communicating directly".
The spokesperson said China hopes North Korea - officially the Democratic People's Republic of Korea - and the USA would treasure the positive progress achieved during the Korean Peninsula talks and continue working to resolve mutual concerns through dialogue and consultation.
Photos released by South Korea's presidential office showed Moon arriving at the North Korean side of the Panmunjom truce village and shaking hands with Kim's sister, Kim Yo Jong, before sitting down with Kim for their summit. KCNA said the leaders also agreed to meet frequently.
The agency said the two leaders had reached a "satisfactory consensus" and expressed "their stand to make joint efforts for the denuclearization of the Korean peninsula".