Mr. Trump withdrew from a planned June 12 Singapore summit with Kim last Thursday, but quickly announced it could still happen.
Trump subsequently wrote on Twitter that the two countries were having "very productive talks" and added that if the summit happens, it will likely still be in Singapore on the initially scheduled date, and could be extended if necessary.
In this handout image provided by South Korean Presidential Blue House, South Korean President Moon Jae-in (Right) hugs with North Korean leader Kim Jong Un (Left) before their meeting on Saturday.
The talks are expected to continue tomorrow at Tongilgak, or "Unification House", the building in the northern part of the demilitarised zone where Kim met Moon on Saturday, The Post reported.
Moon's office said a press conference is set for Sunday morning in Seoul.
Kim's sister Kim Yo Jong also attended the meeting, the Blue House confirmed.
If the summit plans follows through, it would be the first time a sitting United States president and North Korean leader would meet.
North Korea issued a statement Friday saying it is still "willing to give the U.S. time and opportunities" to reconsider talks "at any time, at any format".
Trump-Kim summit: Koreans 'see peace being taken away'
"The Failing @nytimes quotes 'a senior White House official, ' who doesn't exist, as saying 'even if the meeting were reinstated, holding it on June 12 would be impossible, given the lack of time and the amount of planning needed, '" Trump tweeted.
In a tweet earlier in the day, Trump called the North's response to his pullout from the nuclear talks "very good news" and said that "we will soon see where it will lead".
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.
Whether or not the talks will take place in just over two weeks' time is, frankly, anyone's guess, the BBC's David Willis reports from Washington.
For President Donald Trump, Kim Jong Un (kim jawng oon) may be the deal that got away.
"The discussions are just getting started, so we are still waiting to see how they come out, but depending on their outcome, the president could join President Trump and Chairman Kim in Singapore", the official said.
North Korea hardened its rhetoric toward the USA 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 to Libya.
First Vice-Foreign Minister Kim Kye Gwan called the decision "unexpected" and "regrettable".
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