Trump told reporters at the White House that he received the letter on Monday, the first known correspondence of its kind since the two men's summit in Vietnam in February ended without a deal.
"I just received a beautiful letter from Kim Jong-un," Trump said. "It was a very personal, very warm, very nice letter. I appreciate it."
He did not disclose the contents of the message but added that he thinks North Korea has "tremendous potential" under Kim's leadership.
"We have a very good relationship together," Trump said. "Now I can confirm it because of the letter I got yesterday. I think that something will happen that's going to be very positive."
Trump and Kim have exchanged multiple letters since the two sides began to engage diplomatically over North Korea's nuclear weapons program early last year.
Wednesday will mark one year since the two leaders held an historic first summit between the two countries in Singapore and agreed to work toward complete denuclearization of the Korean Peninsula in exchange for security guarantees for Pyongyang.
The second summit in February collapsed due to differences over the scope of North Korea's denuclearization and sanctions relief from the US
In May, North Korea launched short-range missiles in apparent protest over the impasse but Trump played them down as a possible attempt to "get attention."
On Tuesday he again characterized them as "very short term, short range" and claimed that Kim has kept his promise not to test nuclear weapons or long-range missiles.
"It's a whole different deal. But he's kept his word to me. That's very important," he said.
Trump also held out the possibility of a third summit. "Well, it could happen, but I want to bring it further down the line," he said.
Speaking hours earlier at a Wall Street Journal conference, US National Security Advisor John Bolton said a third summit is "entirely possible."
"Really, Kim Jong-un holds the key," he said. "We're ready when they are. So it's anytime that they want to schedule it."
Meanwhile, Trump denied having any information on news reports that Kim Jong-un's half-brother, Kim Jong-nam, was a source for the US Central Intelligence Agency before his assassination at a Malaysia airport in February 2017.
"I don't know anything about that," he said. "I know this, that the relationship is such that that wouldn't happen under my auspices, but I don't know about that. Nobody knows." (Yonhap)