The White House said on Thursday that President Donald Trump received a new letter from North Korean leader Kim Jong Un and responded quickly with a letter of his own.

The correspondence, following up on their Singapore summit, came amid fresh concerns over Pyongyang's commitment to denuclearisation.

Trump early Thursday tweeted his thanks to the North Korean leader “for your nice letter I look forward to seeing you soon!”

The White House did not provide details on the specific content of the letter from Kim, received on Wednesday, or of Trump's reply.

White House spokeswoman Sarah Huckabee Sanders said the letters addressed their commitment to work toward North Korea's “complete denuclearisation.” She said no second meeting is “locked in” as a follow-up to the Singapore summit in June, but they remain open to discussions.

Trump in his tweet expressed gratitude to Kim “for keeping your word” on the return of the remains of more than 50 American service members killed during the Korean War.

Vice President Mike Pence and US military leaders received the remains in Hawaii during a somber ceremony on Wednesday.

The latest letter from Kim arrived on the heels of concerns over North Korea's ballistic missile program and commitment to denuclearisation.

Senior Trump administration officials have urged patience, cautioning that the process of denuclearising North Korea and removing the threat of its long-range missiles will take time.

Secretary of State Mike Pompeo was headed to an Asian security meeting in Singapore, where a meeting this weekend with North Korea's foreign minister was possible.

Trump has sought to show progress from his June 12 summit with Kim. 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 and missile tests.

“A lot of good things are happening. No tests. No rockets flying. But we'll see what happens,” Trump said.

US officials have been closely watching North Korea's willingness to abandon its nuclear ambitions.

The Washington Post on Tuesday reported that US intelligence officials suspect that North Korea is continuing to build new missiles in the same research facility that manufactured the country's ballistic missiles capable of reaching the United States.

The Post also reported that North Korean officials have talked about how they plan to deceive the US about the size of their arsenal of missiles and nuclear warheads and facilities.