The two leaders when they met in Singapore in June announced an agreement in which Kim reaffirmed his "unwavering" commitment to denuclearize, but little progress on that front has been made in the weeks since.
"The ongoing correspondence between the two leaders is aimed at following up on their meeting in Singapore and advancing the commitments made in the U.S.-DPRK joint statement," White House spokeswoman Sarah Sanders said in a statement, using the acronym for North Korea's official name, Democratic People's Republic of Korea.
Trump tweeted about the letter on Thursday, thanking Kim for returning the remains of some American war dead from the 1950-1953 Korean War.
Trump said in that tweet to Kim that "I look forward to seeing you soon!" But there are no current plans for the two men to have a second summit.
The U.S. president has said repeatedly that the deal he reached with Kim has been positive since North Korea has maintained a freeze on nuclear and missile tests and has begun returning U.S. war dead remains.
”I think it’s going to work out very well,” he told supporters at a rally in Tampa, Florida, on Tuesday night. “No tests, no rockets flying. But we’ll see what happens.”
(Reporting by Roberta Rampton and Steve Holland; Editing by Bernadette Baum and James Dalgleish)