A week after President Donald Trump said he received a new "beautiful letter" from North Korean leader Kim Jong Un, US special representative Stephen Biegun said that Pyongyang's promises to give up nuclear weapons lay at the heart of warming relations.

"We can't make enough progress without meaningful and verifiable steps on denuclearization," Biegun said at the Atlantic Council think tank.

"It's absolutely at the core of this. It's what produced this moment to begin with," he said.

But Biegun said that nuclear progress was not a condition for resuming talks, amid a standstill in diplomacy since a February summit between Kim and Trump -- the second between the two leaders after their historic first meeting in Singapore a year ago.

"That is not a condition, and let me say that it will be met with equal vigor on our part to address in parallel all the other commitments that our two leaders made in Singapore," Biegun said.

Trump has offered a new relationship including support for North Korea's creaky economy if Kim gives up nuclear weapons.

But Trump's aides have insisted on no let-up in economic sanctions until North Korea takes concrete action -- a row that contributed to the failure of the Hanoi summit to seal a deal.

Biegun voiced appreciation for North Korean officials' efforts and downplayed a South Korean newspaper report that his counterpart, Kim Hyok Chol, was executed after the Hanoi summit.

"We don't know, and I would tend to think a little bit of that is overblown, but part of it is driven by the fact that so much that happens in North Korea is opaque to us," he said.