The two leaders met during a summit in June. (Image: PA).
Donald Trump has said he is looking forward to meeting North Korea's leader again soon after receiving a "nice letter".
The US President tweeted his thanks to Kim Jong Un "for your nice letter - I look forward to seeing you soon!".
White House spokeswoman Sarah Sanders said the correspondence is "aimed at following up on their meeting in Singapore" and "advancing the commitments made" in the leaders' joint statement there.
Vice President Mike Pence and US military leaders earlier received the remains of missing US soldiers from the Korean War during a ceremony in Hawaii.
The letter comes amid concerns over North Korea's ballistic missile programme and commitment to denuclearisation following the Singapore summit.
Mr Trump in his tweet expressed gratitude to Kim "for keeping your word" on the return of the remains of more than 50 Americans service members killed during the Korean War.
Secretary of State Mike Pompeo is heading to an Asian security meeting in Singapore, where a meeting this weekend with North Korea's foreign minister is possible.
Mr Trump has sought to show progress from his June 12 summit with Mr Kim in Singapore.
He said during a Tuesday rally in Tampa, Florida, that the US was "doing well" with North Korea and noted the return of detained Americans and Pyongyang's ceasing of nuclear testing or missile tests.
"A lot of good things are happening. No tests. No rockets flying. But we'll see what happens," Mr Trump said.
US officials have been closely watching North Korea's willingness to abandon its nuclear ambitions.