South Korean President Moon Jae-in and US President Donald Trump have agreed to push for an early summit between Trump and North Korean leader Kim Jong-un, and closely coordinate to ensure the upcoming summit leads to an agreement on concrete measures to denuclearize North Korea, Seoul officials said Sunday.

The two leaders had a phone conversation late Saturday to discuss the outcome of Moon's summit with Kim.

Trump hailed the historic summit for moving forward inter-Korean ties and welcomed their reaffirmation of a shared goal of complete denuclearization of the Korean Peninsula.

"The two leaders agreed that South Korea and the United States should continue to closely coordinate so that the planned US-North Korea summit could generate an agreement on concrete measures to realize complete denuclearization," Seoul's presidential office Cheong Wa Dae said.

Moon and Kim held a third inter-Korean summit south of the Military Demarcation Line at the border truce village of Panmunjom on Friday, after years of heightened military tensions.

In the so-called Panmunjom Declaration, they reaffirmed their commitment to the complete denuclearization of the Korean Peninsula. Codifying the denuclearization phrase in the summit declaration was hailed as a key achievement from the third cross-border summit, but skeptics pointed to the absence of detailed procedures to achieve the goal.

During the conversation for 1 hour and 15 minutes starting at 9:15 p.m., Trump said that their agreement to achieve a nuclear-free Korean Peninsula is "very good news not only for the two Koreas and but also the entire world."

Trump also expressed his consent to their agreement to push for a formal end to the Korean War, according to the Seoul presidential office.

Moon said the result of the Panmunjom talks will be the cornerstone for a successful US-North Korea summit. Moon hailed the US leader's "bold decision" for making the two successive talks happen.

The two leaders also agreed the meeting between Trump and Kim should be held as early as possible.

Trump has said the meeting will likely take place in May or early June but said at a campaign rally in Washington, Michigan, on Saturday it could happen over the next three to four weeks.

Moon and Trump exchanged opinions on two or three possible cities to host the summit.

"President Trump said he is looking forward to meeting Kim and expects there will be good results from the talks," Moon's office added.

Separately, the White House said the two leaders stressed the need for complete, verifiable and irreversible denuclearization of North Korea.

"President Trump and President Moon emphasized that a peaceful and prosperous future for North Korea is contingent upon its complete, verifiable and irreversible denuclearization," it said.

Trump and Moon "agreed that the unprecedented pressure applied by the United States, the Republic of Korea and the international community through the global Maximum Pressure Campaign has led to this significant moment," it added.

After the phone talks, Trump tweeted Saturday he had a "long and very good talk" with Moon and things were going very well. He said that the time and location of his meeting with Kim were being set, adding they have now narrowed the possible site of the first-ever US-North Korea summit to two or three countries. (Yonhap)