The two-hour talks, which were not announced ahead of time, marked the latest twist in a dramatic week of back-and-forth surrounding the unprecedented talks between Mr Trump and Mr Kim, which had been set for 12 June in Singapore. 

On Thursday, Washington said it was pulling out of the summit, with Mr Trump citing the “open hostility” of recent comments out of Pyongyang. The president seemingly took exception to a statement by North Korean diplomat Choe Son Hui, who referred to Vice President Mike Pence as a “political dummy” and said it was up to the Americans whether they would “meet us at a meeting room or encounter us at a nuclear-to-nuclear showdown”.

North Korea responded that it was still willing to meet “at any time” and in “any format”, a pronouncement that Mr Trump called “very nice” before tweeting on Friday afternoon that his administration was having “very productive talks” with Pyongyang to reinstate the summit.

A statement from North Korea's state news agency KCNA said Kim expressed "his fixed will" on the possibility of meeting with Mr Trump.

In a tweet on Saturday Mr Trump hit out at a report from The New York Times quoting a Senior White House official saying that if the summit is reinstated it would be “impossible” to hold it on 12 June given the planning required at such short notice. “WRONG” the president wrote, offering no evidence to back up his claim. 

Given the nature of the interactions between the Trump administration and North Korea – which have veered from threats of “fire and fury” by the US president and responses in kind from Pyongyang, to suggestions the two nations could work together – the key could be Mr Moon. 

The South Korean president was voted into office on a policy platform of seeking to engage with the North, and he has followed that through. From the participation of Pyongyang in the Winter Olympics in South Korea earlier this year to the latest meeting, Mr Moon has sought to make himself the mediator in trying to solve the crisis. He has flattered Mr Trump by praising his strong leadership and “maximum pressure” campaign through sanctions as reasons Mr Kim has been brought to the negotiating table, but it is obvious that North-South relations are on a surer footing than North Korea-US relations.

The meeting between Mr Moon and Mr Kim on Saturday was the second between the two leaders, with both having taken place in the last few weeks. The first inter-Korean summit in more than a decade took place at Panmunjom border village on 27 April, when both sides declared an aspiration to work towards a nuclear-free Korean peninsula and formally end the 1950-53 Korean War.

It is the North’s nuclear programme that has been the focus of the US administration, with more than 20 missile tests from Pyongyang in 2017 highlighting the progress the nation has been making towards a nuclear warhead-tipped rocket that could reach the US. Indeed, amid months of escalating rhetoric between Mr Trump and Mr Kim’s regime last month, Pyongyang threatened to fire missiles at the US territory of Guam.

Saturday’s talks again took place in Panmunjom, although it is not yet known how long the meeting lasted or who had asked for it.

“The two leaders candidly exchanged views about making the North Korea-US summit a successful one and about implementing the Panmunjom Declaration,” South Korea’s presidential spokesman said in a statement.

Mr Moon, who returned to Seoul on Thursday morning after meeting Mr Trump in Washington earlier in the week in a bid to keep the US-North summit on track, is due to announce details of the meeting with Mr Kim early on Sunday.

One of the photos released by the presidential Blue House on Saturday showed Mr Moon and Mr Kim hugging each other after their meeting at Tongilgak, the North’s building in the truce village. The previous summit was held at the southern side of the border.

They were accompanied by South Korean intelligence chief Suh Hoon and his North Korean counterpart Kim Yong Chol, who is in charge of inter-Korean affairs. Another photo showed Moon shaking hands with Kim Jong-un’s sister, Kim Yo-jong.

Mr Moon is walking a political tightrope, with decades of difficult diplomacy between world powers and the North littered with examples that show the situation can change very quickly – even without the often impulsive input of Mr Trump.

South Korea was caught off guard by Mr Trump’s abrupt cancellation of the Singapore summit. Mr Moon said the US president’s decision left him “perplexed” and was “very regrettable”. Mr Moon’s push to keep the 12 June summit on track may also be driven – at least in part – by the worry of a US president who thinks less of the traditional alliance with Seoul than his White House predecessors.

A number of analysts have highlighted what they see as the risky nature of Mr Trump’s cancellation of the Singapore meeting and how it could play into Mr Kim’s hands in terms of propaganda value. Adam Mount, a senior fellow at the Federation of American Scientists and director of its Defense Posture Project, which looks at deterrence and the use of nuclear weapons, said that Mr Moon’s latest meeting with Mr Kim was a “clear demonstration of how dangerous Trump’s temper tantrum was”.

“When Kim Jong-un was allowed to split the negotiations into separate tracks with Trump and Moon, he gained leverage over both,” Mr Mount tweeted. “Moon was sitting too alone at the table today, without the full weight of the United States.”

However, the White House sought on Saturday to make clear it was still preparing for the summit, with reports of an advance team of around 30 White House and State Department staff preparing to leave for Singapore over the weekend.

In a statement, White House Press Secretary Sarah Sanders said: “The White House pre-advance team for Singapore will leave as scheduled in order to prepare should the summit take place.” 

On Friday, Mr Trump was asked if North Korea was playing games with the US over the status of the summit, and he replied that “everyone plays games”. But for Mr Moon the stakes are much higher.

“Moon Jae-in is not playing a game: he must keep his people safe from war,” said Mr Mount. “Each of Trump’s whims shakes the walls of the Blue House.”

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