But North Korea responded later that evening expressing a willingness to talk to the U.S.at any point.
President Moon is greeted by Kim Yo-jong, the younger sister of Kim Jong-un, on his way into Tongilgak for the summit meeting.
Trump told reporters at the White House that both sides would like the summit to happen. Trump, speaking after welcoming an American who had just returned to the US after being freed from jail in Venezuela, said that working level meetings with the North are ongoing at a location "not so far away from here" and they have "gone very very well".
"We're doing very well in terms of the summit with North Korea", Trump said at the White House.
North Korea's vice foreign minister said that they are willing to meet any time to resolve any issues.
President Donald Trump speaks during a graduation and commissioning ceremony at the U.S. Naval Academy, May 25, 2018, in Annapolis, Md.
Trump earlier on Friday hailed the quote "warm and productive statement" from Pyongyang...
He did not detail the nature of the new US communication with the North.
Defense Secretary Jim Mattis said Friday that the meeting may still take place if diplomats can pull it off.
Analysts say Kim's diplomatic outreach in recent months after a flurry of nuclear and missile tests in 2017 indicates he is eager for sanctions relief to build his economy and the worldwide legitimacy the summit with Trump would provide.
"Moon Jae-in is acting decisively to keep his people safe from war", said Adam Mount, a nuclear expert at the Federation of American Scientists.
"No, we'll see what happens, it could even be the twelfth".
White House spokeswoman Sarah Sanders said a team of White House and US State Department officials would leave for Singapore on schedule this weekend for the planned 12 June summit.
Ms Choe, who has been involved in several diplomatic interactions with the USA over the past decade, said the North would not "beg" for dialogue and warned of a "nuclear showdown" if diplomacy failed. "There had to be something to come out of it".
He said, if the summit does happen it will likely remain in Singapore and be on that same June 12 date.
Kick-starting talks, however, will face a number of obstacles, including a possible credibility gap created by the dramatic about-face.
Meeting at a border truce village, Kim and Moon discussed carrying out the peace commitments they reached in their first summit, as well as Kim's potential meeting with Trump, Moon's office said. The two officials signed a joint declaration, agreeing to take measures to support global efforts aimed at denuclearization of the Korean Peninsula and proceed with reunion programs for the separated Korean families.
While details of what became the fourth inter-Korean summit have not been announced, it was nearly certainly about the Kim-Trump summit scheduled for June 12 in Singapore before Trump called it off Thursday.
Moon will announce the outcome of the meeting at 10 a.m. Sunday, according to Yoon, as part of an agreement with the North.
Earlier on Friday, speaking to reporters outside the White House in Washington, the United States president indicated that the summit could still be salvaged, saying: "We're gonna see what happens".
Bruce Klingner, a former Central Intelligence Agency division chief for the Koreas who is now at the conservative Heritage Foundation, said the latest developments aren't so much a PR campaign as a negotiating tactic.
Walking past North Korean soldiers saluting him, Moon met Kim in the lobby of Tongilgak, their second up-close-and-personal encounter in 29 days.
And given Trump's entertainment background, other experts said it would be wise not to attempt to divine the machinations of the mercurial president until the dust settles.
Kim ordered the complex to be finished by April 15 next year to mark the birthday of his late grandfather and North Korea founder Kim Il Sung.
And Mr Trump added: "Everybody plays games". "Tune in at the end...or next season", he wrote.