The US says it is already looking at scheduling a new meeting.
White House spokeswoman Sarah Sanders said the “very warm” letter showed Pyongyang’s “continued commitment to focus on denuclearisation”.
Negotiations on the topic appeared to have stalled after the two leaders’ historic summit in Singapore in June.
“The primary purpose of the letter was to request and look to schedule another meeting with the president, which we are open to and are already in the process of co-ordinating that,” Ms Sanders said.
She gave no indication of when a second meeting between the two leaders could potentially take place.
South Korean President Moon Jae-in welcomed the news, saying that “complete denuclearisation of the Korean peninsula is an issue that should fundamentally be resolved between the US and North Korea through negotiation”,
Mr Moon had been crucial in brokering the Singapore summit in June and is himself scheduled to meet with Mr Kim in Pyongyang next week for a third round of face-to-face talks.
The BBC’s Laura Bicker in Seoul says Mr Moon sees himself as a mediator between the two sides and has called on them both to make bold steps.
Pyongyang’s letter comes a day after Yukiya Amano, the head of the UN’s nuclear watchdog, warned that North Korea’s ongoing nuclear activities violated UN Security Council resolutions.
International inspectors are banned from North Korea, but Mr Amano said they were ready to return if a political agreement could be reached.
‘Very warm, very positive’
Ms Sanders also praised North Korea’s military parade last weekend, saying that it “for once was not about their nuclear arsenal”. She attributed that to the “tremendous success” of Mr Trump’s policies.
While North Korea did roll out soldiers, tanks and other weaponry, the parade did not display any intercontinental ballistic missiles (ICBMs) at the parade marking its 70th anniversary, reports say.
A display of ICBMs – which can reach the US mainland, potentially carrying a nuclear warhead – would have been seen as provocative.
Mr Trump himself thanked the North Korean leader via Twitter, saying the parade was “a big and very positive statement from North Korea”.
“Thank you to Chairman Kim. We will both prove everyone wrong!”
At their June summit in Singapore, the two leaders signed an agreement to work towards the denuclearisation of the Korean peninsula.
But it did not include a timeline, details or any mechanisms to verify the process.
High-level talks and visits have continued, but the most recent scheduled trip by US Secretary of State Mike Pompeo was called off at the last minute.
Both sides have also blamed each other for stalling negotiations while insisting that they were committed to the process.