In a press briefing, White House spokesperson Sarah Sanders said "we have responded to Chairman Kim's letter, the president has, and that letter will be delivered shortly," Xinhua reported.

Refusing to reveal the contents of the two letters, Sanders only said that the letters "addressed their commitment from their joint statement" made in the Singapore summit.

"They are going to continue to work together towards complete and total denuclearization," she said.

Trump earlier tweeted that he looks forward to seeing Kim "soon."

However, Sanders said that although the White House is "certainly open to that discussion," there is not "a second meeting that is currently locked in or finalized."

Speaking of North Korea's repatriation of the remains of fallen US soldiers from the 1950-1953 Korean War, Sanders said that the US side was "incredibly grateful and thankful."

"We are going to continue to work with North Korea," she said, referring to North Korea.

Earlier on Thursday, Sanders said in a statement that Trump has received a letter from Kim, and "the ongoing correspondence between the two leaders is aimed at following up on their meeting in Singapore and advancing the commitments made in the US-North Korea joint statement."

Trump also thanked Kim via Twitter for returning the remains of fallen US soldiers from the Korean War.

The repatriation of the remains of US service members was part of the commitment made by Trump and Kim in their Singapore summit on June 12. A ceremony was held on Wednesday in Hawaii to receive the remains.