Earlier Thursday, a diplomatic source here said the US and North Korea were still in talks to set a date for working-level dialogue over the repatriation of remains of American troops.
Addressing Mr Trump, Kim describes their June 12 summit in Singapore, and the resulting joint statement, as the "start of a meaningful journey".
"I firmly believe that the strong will, honest efforts and unique approach of myself and Your Excellency Mr President aimed at opening up a new future between the DPRK and the USA will surely come to fruition", Mr Kim wrote, referring to North Korea by its acronym.
The North Korean leader affirmed his firm belief that "the strong will, honest efforts and unique approach" the two have "aimed at opening up a new future between the DPRK and the USA will surely come to fruition".
Kim also voices hope that "the invariable trust and confidence in Your Excellency Mr. President will be further strengthened in the future process of taking practical actions".
In the tweet sent on Thursday, Trump declared "Great progress being made!" with North Korea, despite signs that diplomacy between the traditional adversaries was running into trouble.
Mr Kim agreed in broad terms to "work toward denuclearisation" of the Korean Peninsula when he met Mr Trump in Singapore last month.More news: British woman Dawn Sturgess exposed to Novichok nerve agent dies
An estimated 30,000 United States soldiers died in the Korean war.
Last week, North Korea accused the United States of using a "gangster-like" tactics to push it towards nuclear disarmament after a fresh round of high-level talks. The Trump administration released a letter from Kim, dated July 6, after officials say it was given to Pompeo to deliver to Trump when his delegation was recently in the DPRK.
North Korea, meanwhile, lashed out at Washington for presenting "gangster-like demands" for complete, verifiable and irreversible denuclearization.
In another development on Thursday, North Korean officials failed to turn up for talks with a USA team on the repatriation of the remains of American soldiers killed in the 1950-53 Korean war.
North Korea has offered to rearrange the meeting over war remains for Sunday instead, US State Department spokeswoman Heather Nauert said.
Earlier Thursday, after attending the NATO summit in Brussels, he told reporters that there were clear signs of progress with North Korea, most notably that no missile and nuclear tests have occurred for nearly nine months.
"The US wanted to have talks as early as 12 July but the North did not seem to be ready", an unnamed Seoul official said. -North Korea relations worsened over North Korea's nuclear weapons program.