The pair agreed to restart working-level dialogue during an impromptu meeting at the Demilitarised Zone dividing North and South Korea in June, but those talks have yet to begin.
"We are willing to sit face-to-face with the US around late September at a time and place that we can agree on," Choe Son-hui, the North's vice foreign minister, said in a statement carried by the official Korean Central News Agency.
Choe's comment followed his warning in late August that the North's "expectations of dialogue with the US are gradually disappearing", after Pyongyang conducted a series of weapons tests to protest joint US-South Korean military exercises.
|North Korea's vice foreign minister Choe Son-hui (Yonhap)|
Asked about the proposal for lower-level September talks, Trump told reporters: "I have a very good relationship with Chairman Kim. I always say having meetings is a good thing. We'll see what happens."
The period suggested by North Korea would correspond with the United Nations General Assembly in New York.
On Monday, Choe recalled Kim's comments that the North would wait until the end of the year for Washington to "quit its current calculation method".
Choe repeated Pyongyang's call for the US to come up with an "acceptable calculation" or risk jeopardising the entire diplomatic process.
"I think the US has since had enough time to find the calculation method that it can share with us," he said.
- Nuclear warhead development -
Kim and Trump adopted a vaguely-worded statement on the "complete denuclearisation of the Korean peninsula" at their first summit in Singapore in June last year, but little progress has since been made on dismantling the North's nuclear programme.
A report published by a United Nations panel of experts last week said the North's development of nuclear warheads has not stopped, despite a moratorium it declared on nuclear blasts and long-range missile launches.
Last year, the North also said its Punggye-ri atomic bomb test site would be dismantled as it was no longer needed.
With its recent weapons tests, North Korea has demonstrated it can "master key components of ballistic missile systems, including solid-rocket fuel production... and the capacity to penetrate ballistic missile defence systems", the UN report said, citing an opinion of a member state.
US officials have called these short-range missile launches provocations, although Trump himself has avoided criticising them.
North Korea is under heavy US and UN sanctions over its nuclear programme, and has criticised Washington's position that sanctions against the isolated regime will not be lifted until the country gives up its nuclear weapons.
On Friday, the US special envoy to North Korea, Stephen Biegun, said Pyongyang must stop blocking nuclear talks.
"If we are to succeed, North Korea must set aside its search for obstacles to negotiations and instead seek the opportunities for engagement while that opportunity lasts," he said.
"We have made clear to North Korea that we are prepared to engage as soon as we hear from them. We are ready, but we cannot do this by ourselves," Biegun said in a speech.
When asked by AFP about the latest North Korean offer, a State Department official replied: "We don't have any meetings to announce at this time."
Richard Johnson, of the Washington-based think tank Nuclear Threat Initiative, said if the talks are serious, North Korea would need to send "subject matter experts on nuclear, economics, sanctions, peace, legal issues," alongside officials like Choe.
"At a bare minimum, the NK team will need to come with a mandate to negotiate -- real gives and takes," Johnson said on Twitter, adding the US would also need to show flexibility. (AFP)