North Korean and US officials are holding behind-the-scenes talks to set up a third summit between Donald Trump and Kim Jong Un on the fate of the North’s expanding nuclear arsenal, according to South Korea’s president.

Moon Jae-in was speaking four months after a second meeting between the leaders in Vietnam collapsed without agreement.

There have been no public meetings between Washington and Pyongyang since the breakdown of the Hanoi summit, but the prospects for a resumption of diplomacy have brightened since the two men recently exchanged personal letters.

The US president called Mr Kim’s letter “beautiful” while the North Korean leader described Mr Trump’s as “excellent”, although the contents of the letters have not been disclosed.

Kim Jong Un and Donald Trump in Hanoi
Kim Jong Un and Donald Trump in Hanoi (Evan Vucci/AP)

Mr Moon said the two men’s “willingness to engage in dialogue has never faded” and their recent letter exchanges prove that.

The South Korean leader, a liberal who met Mr Kim three times last year, has made dialogue with the North as a route to forging peace on the Korean peninsula a centrepiece of his presidency.

He said he does not see the Hanoi summit as a failure as it served as a chance for Washington and Pyongyang to better understand each other’s positions and “put everything they want on the negotiating table”.

“The success of denuclearisation and the peace process on the Korean peninsula cannot be determined by a summit or two,” Mr Moon said, adding that the discussions in Hanoi will form the basis for future talks. “Both sides clearly understand the necessity for dialogue,” he added.

Despite the deadlocked negotiations, Mr Trump and Mr Kim have described their personal relationship as good. Asked whether Mr Kim’s recent letter included a mention of another summit, Mr Trump said: “Maybe there was.”

“At some point, we’ll do that,” he told reporters at the White House on Tuesday. “Getting along very well. He’s not doing nuclear testing.”

But in another reminder of North Korea’s continued mistrust of the US, its foreign ministry said earlier on Wednesday that it will not surrender to US-led sanctions and accused Washington of trying to “bring us to our knees”.