Trump scrapped the meeting entirely in a letter to Kim on Thursday, after repeated threats by North Korea to pull out over what it saw as confrontational remarks by US officials demanding unilateral disarmament.
Earlier this week President Trump abruptly canceled his plans to meet Kim in a meeting set to take place June 12th in Singapore.
"It was a very nice statement they put out".
The two nations have remained estranged since the Korean War, and tensions between them had risen in recent years as the North sought to advance its nuclear weapons and ballistic missile programs.
"If it does happen, will likely remain in Singapore on the same date, June 12th", he wrote, adding the meeting could be extended further if necessary.
The two leaders embraced as Moon departed.
But North Korea responded Friday by saying it was willing to talk to the United States "at any time" - a reaction Trump welcomed as "warm and productive". Trump rapidly tweeted that the statement was "very good news" and told reporters that "we're talking to them now".
Trump views the meeting as a legacy-defining opportunity and has relished the press attention and the speculation about a possible Nobel Peace Prize.
Trump told reporters that he had no hard feelings toward North Korea because "everybody plays games". "You know that better than anybody".
Those included the North Korean decision to cancel a planned high-level meeting with the South to protest joint military exercises held annually by Seoul and Washington, the official said, speaking on condition of anonymity.
On Saturday, White House spokeswoman Sarah Huckabee Sanders said a team will leave for Singapore as scheduled "to prepare should the summit take place".
In a tweet Friday, Trump maintained "very productive talks" were continuing on the North Korean summit.
United States officials have talked about a comprehensive one-shot deal in which North Korea fully eliminates its nukes first and receives rewards later.
North Korea hardened its rhetoric toward the United States on Thursday, lashing out after remarks by Vice President Mike Pence and the White House national security adviser, John Bolton, that had linked the country with Libya.
Activists march toward the Unification Bridge, which leads to Panmunjom in the Demilitarized Zone, during the 2018 DMZ Women Peace Walk in Paju, South Korea, Saturday, May 26, 2018. Under the Trump administration, the Central Intelligence Agency, where now-Secretary of State Mike Pompeo served as director, has taken an unusually prominent role in back-channel negotiations.
In Pyongyang, North Korean Vice Foreign Minister Kim Kye Gwan said North Korea's criticisms of certain US officials had been a reaction to American rhetoric and that the current antagonism showed "the urgent necessity" for the summit.
However, the North quickly became angered by comments from senior U.S. officials who made comparisons with Libya.
Moon used his hosting role for the Winter Olympics in February to renew the push to resolve the nuclear standoff with Pyongyang.
In a letter to Kim announcing the cancellation, Trump objected specifically to a statement from senior North Korean diplomat Choe Son Hui.
USA officials have said they would like to see the complete and irreversible dismantling of the North's nuclear programme and weapons, allowing worldwide inspectors to check every step of that process, before any talk of sanctions being lifted.
The two leaders "exchanged their opinions" on among other things successfully carrying out a future US-North Korea summit, according to the statement.