Trump tweeted on Thursday alongside a copy of the letter, dated July 6.
"I firmly believe that the strong will, honest efforts and unique approach.aimed at opening up a new future between the DPRK and the U.S. will surely come to fruition".
While the summit represented a major thaw in relations, US sanctions on North Korea, among the most stringent in the world, will remain in place until the Kim regime implements "final, fully verified denuclearization", US State Secretary Mike Pompeo said last week.
The letter however made no specific mention of the efforts toward denuclearization.
"I extend my conviction that the epochal progress in promoting the DPRK-US relations will bring our next meeting forward".
Mr Kim agreed in broad terms to "work toward denuclearisation" of the Korean Peninsula when he met Mr Trump in Singapore last month.
One observer pointed out that Trump would have likely criticized his response to North Korea had it been the reaction of another president.More news: 8th boy rescued from Thai cave, 5 people remain
"Rocket Man" is a nickname Trump used to mock Kim past year when the two traded insults over North Korea's testing of nuclear weapons and ballistic missiles targeted at the US.
The top Democrat on the Senate Intelligence Committee has asked the head of USA intelligence to assess whether North Korea's posture and actions regarding denuclearization match up with the Trump administration's claims. "We had anticipated the USA side would come with a constructive idea, thinking we would take something in return", a foreign ministry official said in a statement. A confidential United States report presented to the UN security council, however, estimated the delivery of at least 759,793 barrels of oil products to North Korea between 1 January 1 and 30 May.
Mr. Pompeo denied that the talks went poorly, telling pool reporters that progress was made "in every element of our discussions".
US State Department spokeswoman Heather Nauert said North Korea has now offered to meet the delegation on Sunday.
The remains of some United States soldiers were last returned in 2007, when then-New Mexico Governor Bill Richardson visited Pyongyang.
"Repatriation of USA remains from North Korea will be important not just to American families, but to building trust between the US and North Korea as well", Pompeo said earlier this week.