Trump said the lines of communication had opened between the two countries after Trump administration officials yesterday said that the North Koreans had become unresponsive, leading to the planned summit's cancellation.
North Korea said Friday it is still willing to sit for denuclearization talks with the United States "at any time, (in) any format". He did not confirm how the secret meeting was arranged or which side asked for it.
He also said holding the summit on June 12 in Singapore is still a possibility. Moon was accompanied by South Korea's national intelligence service chief, Suh Hoon. Another photo showed Moon shaking hands with Kim Jong Un's sister, Kim Yo Jong.
The White House said Trump dictated the letter himself.
At the Pentagon, Defense Secretary Jim Mattis said Friday that "the diplomats are still at work on the summit" but declined to say whether he thought the event would take place on June 12.
Trump said in a tweet Friday that the United States is "having very productive talks with North Korea" about the summit.
Relations between the U.S. and North Korean became progressively optimistic over the past few months, after Kim Jong-un invited Mr Trump for a meeting in March. (This is not an unrealistic scenario.) Despite Trump's hints that the summit could still happen, that should make any possible deal between Kim and Trump a non-starter for us, right?
"We find it fortunate that the embers of the North Korea-US talks are reignited".
"It was a very nice statement they put out".
South Korea's presidential spokesman said in response: "We are cautiously optimistic that hope is still alive for US-North Korea dialogue".
He cancelled the summit on Thursday, blaming the North's "open hostility".
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Kim Kye Gwan, first vice minister of foreign affairs, issued a statement earlier on Friday saying North Korea was "willing to give the US time and opportunities" to reconsider talks.
After that, a senior White House official said the North had reneged on its promises ahead of the summit.
Trump welcomed the statement as "warm and productive".
Trump canceled the meeting a day earlier, citing "tremendous anger and open hostility" from North Korea.
Pyongyang on Friday said in a statement that it is still willing to "sit with the US side" for talks "at any time". If they do that and Kim agrees to stop testing his nukes (while still keeping the ones he has), then China is going to breathe a huge sigh of relief and probably go back to fully and openly supporting North Korea. The plan followed months of war threats and insults from both leaders.
Comments in North Korea's state media indicate Kim sees any meeting with Trump as an arms control negotiation between nuclear states, rather than a process to surrender his nukes.
Trump's latest about-face sent officials scrambling in Washington. Moon urged direct talks between Trump and Kim to get things back on track.
The leaders of North and South Korea have met in the demilitarised border area between the two countries.
North Korea responded to the cancellation in a strikingly conciliatory tone.
Politico magazine reported that an advance team of 30 White House and US State Department officials was preparing to leave for Singapore later this weekend.