The two leaders travelled huge distances to Hanoi for a second summit over dismantling Pyongyang's nuclear programme but the two-day meeting broke down suddenly with the two sides failing to agree on a joint statement as scheduled.

A senior administration official, who requested anonymity, said the North Koreans wanted "many, many billions of dollars in sanctions relief" but were "unwilling to impose a complete freeze on their weapons of mass destruction programmes".

Lifting the sanctions "would in effect put us in a position of subsidising the ongoing development of weapons of mass destruction in North Korea," the official added. "The weapons themselves need to be on the table."

"The President in his discussions challenged the North Koreans to go bigger. The president encouraged Chairman Kim to go all in. And we were going to -- we were prepared to go all in as well," he said.

North Korea's foreign minister convened journalists from a handful of countries for a surprise midnight press conference in Hanoi to present Pyongyang's position that it had only requested partial sanctions relief.

In return -- and in what the minister Ri Yong Ho called a "realistic proposal" -- the North had offered to "permanently and completely dismantle all the nuclear production facilities in the Yongbyon area, in the presence of US experts".

But one of the problems was a precise definition of what is located at Yongbyon -- an enormous complex containing "more than 300 different separate facilities", the US official noted.

"What the North Koreans proposed to us was closing down a portion of the Yongbyon complex."

A further sticking point was the ever-thorny issue of an exact definition of "denuclearisation".

While Washington wants North Korea to give up its nuclear weapons programme, Pyongyang sees denuclearisation more broadly.

The North's view includes an end to sanctions and what it sees as US threats -- usually including the American military presence in South Korea, and sometimes in the wider region.

The official said the definition was extensively discussed at working-group level but "they haven't agreed to it".

Despite the chasms still separating the two sides, the official -- like his boss Trump -- sought to strike a positive tone.

"We're actually encouraged by where we're going," he said."

"We didn't get close enough at this summit but we're encouraged by the opportunities ahead of us."