"He's calm. I'm calm," Trump said this week.

The US leader has claimed a diplomatic win, citing the "tremendous progress" and "great responses" from Pyongyang since he met Kim at a summit in Singapore in June.

The historic meeting resulted in a lofty joint statement, though momentum appeared to have stalled in recent weeks, with the two sides unable to agree on how Kim would get rid of his nuclear weapons.

But on Friday, US Secretary of State Mike Pompeo said he hoped for a second summit between Trump and Kim "before too long" to press ahead with efforts to denuclearize North Korea.

"I hope the two leaders get together again to continue to make progress on this incredibly important issue for the entire world," Pompeo said.

The development came after Kim this week welcomed South Korean President Moon Jae-in to Pyongyang.

The North Korean leader agreed to close a missile testing site, giving new momentum to nuclear negotiations with the US, and said the North could dismantle its best-known nuclear facility at Yongbyon -- provided the US takes "corresponding measures."

Experts praised the positive developments, but cautioned much remains to be done, noting the steps Kim has so far taken do not ensure North Korea will get rid of its nuclear weapons.

"Trump still thinks the first (summit) was a major success," Jon Wolfsthal, director of the Nuclear Crisis Group, told AFP.

"He needs a foreign policy success soon to reverse his domestic decline," the former Obama administration official added.

- Denuclearization first -

Pompeo has said Washington is ready to "immediately" begin negotiating with Pyongyang, and has invited his North Korean counterpart to meet on the sidelines of next week's United Nations General Assembly in New York.

But the top US diplomat has quietly ignored North Korea's demand for countermeasures, such as sanctions relief, as the US has long insisted on denuclearization before it rewards Pyongyang.

"Nothing can happen in the absence of denuclearization," State Department spokeswoman Heather Nauert said.

"Denuclearization has to come first," she added. "In order to get to the denuclearization of North Korea, sanctions must be enforced. We cannot let our foot off the gas."

While Washington isn't budging on sanctions, Trump has indicated he is ready to independently offer other concessions, a move Kim may be counting on.

"After a nice letter from (Kim), the president says he's willing to have a summit -- though none of the problems have been fixed," Bruce Klingner, who formerly worked for the CIA and now specializes in Korean and Japanese affairs for the Heritage Foundation think tank, told AFP.

"Pyongyang is trying to decouple Trump from the rest of the administration. They see him as more likely to offer concessions, as he did in Singapore, without demanding things in return -- and Kim is right to think so."

The US could try to use North Korea's eagerness to meet with Trump as some sort of leverage to get concrete steps ahead of the meeting," Klingner added.

Still, many experts view the prospect of new talks positively.

While the announcements made by the North this week are limited, "it is the first time where we have seen this initiative where North Koreans are willing to talk to South Koreans about denuclearization," said Joseph Yun, who was the US special representative for North Korea until days before Trump accepted the Singapore summit invitation.

"The offer is worth pursuing," added Wolfsthal.

Contrary to the all-or-nothing approach espoused by the Trump administration, US officials appear to be engaging in a process of limited reciprocal concessions.

The Singapore summit agreements imply "action for action rather than one party doing everything at once," Yun said.

"Any expectation that North Korea will completely denuclearize, give up nuclear weapons, give up missiles, give up fissile material, destroy sites before anything is done on any other front is completely unrealistic," he added.