Secretary of State Mike Pompeo was ensconced in an elegant Pyongyang guest house for a second day of talks with North Korean leader Kim Jong Un's right-hand man Kim Yong Chol.

"Our policy hasn't changed," Pompeo's spokeswoman Heather Nauert told reporters.

"Our expectation is exactly what the president and Kim Jong Un jointly agreed to in Singapore, and that is the denuclearisation of North Korea," she said.

The talks were held at a large villa inside an official compound in Pyongyang, close to the imposing mausoleum where North Korea's former helmsmen Kim Il Sung and Kim Jong Il -- the current leader's grandfather and father -- lie in state.

As the day began, Pompeo left the compound to go to a location where he could make a secure call to President Donald Trump away from potential surveillance, then returned to restart talks at around 9:00am (0000 GMT).

Kim Yong Chol asked Pompeo if he had slept well on his first overnight stay in the country, adding: "But we did have very serious discussion on very important matters yesterday.

"So thinking about those discussions you might have not slept well last night," he suggested.

Pompeo responded that he had slept "just fine" but the exchange suggested tougher talks ahead.

Pompeo warned that "the path toward complete denuclearisation building a relationship between our two countries is vital for a brighter North Korea and the success that our two presidents demand of us."

Kim replied: "Of course it is important. There are things that I have to clarify."

"There are things that I have to clarify as well," Pompeo responded.

- Crisis over? -

Pompeo, who is on his third visit to Pyongyang, began the outreach when he was still Trump's CIA director and remained the pointman on negotiations after the process became public and he became secretary of state.

In comparison to past international nuclear disarmament negotiations, the discussions between Washington and North Korea on thawing ties and dismantling the North's arsenal appear to be proceeding in reverse.

Last month, Kim and Trump met in Singapore and signed a statement committing Pyongyang to "work towards complete denuclearisation of the Korean Peninsula" in exchange for US "security guarantees" and peace in the decades-old stand-off.

But rather than the two leaders crowning years of detailed negotiation with their one-on-one meeting, the short statement marked instead the start of a diplomatic long slog, and Trump earned the scorn of Korea watchers and non-proliferation experts when he declared the crisis over.

- 'Nitty gritty details' -

The task of establishing the disarmament programme now falls to Pompeo, who is seeking a formal declaration by the North of the size of its nuclear programme as well as an eventual timetable for it to be stood down under international verification and inspection.

Many experts doubt Kim's sincerity -- a nuclear deterrent to the US military forces massed in South Korea has long been a strategic goal of his isolated, autocratic regime -- and few expect this to be a quick process, even if Washington wants results within a year.

Pompeo, accompanied by senior State Department and CIA officials, held several hours of talks on Friday evening and had a working dinner with Kim Yong Chol at which both were "cracking jokes" and "exchanging pleasantries," according to Nauert.

They also, she said, agreed to set up working parties to pursue the "nitty gritty details" of how to make good on promises made in Singapore.

They were expected to talk all day Saturday before Pompeo heads to Tokyo to brief Washington's Japanese and South Korean allies.