Seoul, South Korea (CNN) US President Donald Trump seemed to revive hopes for a summit with North Korean leader Kim Jong Un next month, telling reporters at the Oval Office late Saturday, "we're looking at June 12th in Singapore. That hasn't changed."

His remarks came as South Korean President Moon Jae-in delivered his first public comments since a surprise meeting with the North Korean leader on Saturday, saying Kim was still committed to denuclearization.

"What is unclear to Chairman Kim Jong Un is not the will for denuclearization, but the concern that if (North Korea) denuclearizes, whether the US can end hostile relations and guarantee the security of the (Kim) regime," Moon said.

On Thursday, Trump abruptly canceled the highly-anticipated US-North Korea summit, scheduled for June 12, citing hostile comments from top North Korean officials and concern about the country's commitment to giving up its nuclear weapons.

Moon said Sunday that practical talks will be held between the US and North Korea "very soon," and the outcome of those talks will dictate if the June 12 summit will go ahead. He added that he expected the practical talks and summit to go "very smoothly."

Trump said there was a "lot of goodwill" for the meeting. "A lot of people are working on it. It's moving along very nicely."

Experts were more skeptical however. CNN global affairs analyst Joseph Yun said a joint declaration of some sort on June 12 was far more likely, given the short time period to iron out differences before Trump and Kim sit down together.

"Let's be realistic. These summits take months not weeks to prepare," he said.

Many North Korea watchers have previously warned that things have been moving too fast, and urged Washington to hold more preparatory meetings in the run up to a formal summit.

Kim Jong Un and Moon Jae-in talk during a meeting at the northern side of the Panmunjom in North Korea. It was their second face-to-face meeting.

China said it supports a meeting between Trump and Kim.

"We hope to see the summit take place as scheduled, and be held successfully to bring peace to the Peninsula and good news for the rest of the world," Chinese Foreign Ministry spokesman Lu Kang told CNN.

Surprise meeting

The South Korean President met Kim on the North side of the demilitarized zone (DMZ) between the two Koreas late Saturday. Their last meeting on April 27 was held on the South side.

During his meeting with Kim, Moon said the North Korean leader "once again has made clear his will for the complete denuclearization of the Korean Peninsula and expressed his intent to settle the history of war and confrontation, and to cooperate for peace and prosperity through the success of the North Korea-US summit."

Moon and Kim embrace before their meeting on Saturday.

Moon said on Sunday: "Our journey to the complete denuclearization of the Korean Peninsula and a permanent peace agreement cannot be stopped, and we have decided to cooperate closely."

Moon also suggested a US-North Korean hotline could be opened in future, and possibly a three-way hotline with South Korea, though he said a trilateral summit should be held first.

"I hope if the NK-US summit is successful, through the three-way summit, we can declare the end of the war," Moon said.

In a statement Sunday released before Moon spoke, North Korean state media said the meeting took place "all of a sudden."

North Korea informed its own people of the meeting between Moon and Kim, with reports in state-run media hailing a "new era of reconciliation and unity."

Moon apologized to South Korean media Sunday for the delay in releasing details about the meeting, saying he had done so at the North Korean leader's request, in order to let Kim tell his country first.

"Panmunjom, the historical land which had once come under global spotlights as a symbol of peace for making a new start of the north-south relations and opening up a new era of reconciliation and unity, witnessed the significant meeting between the top leaders of the north and the south once again after 29 days," North Korea's Central News Agency (KCNA) said.

South Korean President Moon Jae-in and North Korean leader Kim Jong Un seen on the front page of the Rodong Sinmun after a surprise second meeting on May 26, 2018.

KCNA said Moon signed the visitors' book in Thongil House where the meeting took place, writing ""Peace and Prosperity of the Korean Peninsula, together with Chairman Kim Jong Un."

"At the talks there were in-depth exchanges of opinions to tackle the matters which should be resolved to quickly carry out the Panmunjom Declaration agreed upon at the third North-South summit and to realize the denuclearization of the Korean peninsula and achieve regional peace, stability and prosperity," KCNA said.

The Panmunjom Declaration was the result of the first meeting between Moon and Kim in April, and committed the two countries to denuclearization of the Peninsula and talks to bring a formal end to the Korean War.

Kim thanked Moon "for much effort made by him" towards the US-North Korea summit, and said he hoped to improve relations with Washington and "establish mechanism for permanent and durable peace."

Photos of Kim and Moon embracing and shaking hands also appeared on the Sunday front page of state-run newspaper Rodong Sinmun.

Singapore summit called off

In a tweet Friday, Trump had said "very productive talks" were continuing on the North Korean summit.

"We are having very productive talks with North Korea about reinstating the Summit which, if it does happen, will likely remain in Singapore on the same date, June 12th., and, if necessary, will be extended beyond that date," Trump wrote.

Moon, who has been a mediator between Trump and Kim, called an emergency meeting in the middle of the night Thursday after Trump called off the June summit in a letter to Kim.

North Korea had indicated it wasn't happy with the way things were progressing when it called of high level talks with South Korea over joint US-South Korean military drills.

JUST WATCHED CNN gets access to US-S. Korea military drills Replay More Videos ... MUST WATCH CNN gets access to US-S. Korea military drills 02:04

for Trump came when a North Korean Foreign Ministry official US officials said the final strawfor Trump came when a North Korean Foreign Ministry official called Vice President Mike Pence a "political dummy" and hinted that Pyongyang was ready for a "nuclear showdown" should diplomacy fail.

The US also claimed in a background briefing that North Korean officials had failed to show up in Singapore to arrange logistics for the June meeting.

But a North Korean Foreign Ministry official said Friday that Kim was still willing to meet with Trump "at any time."