"I think we'll have another meeting," Efe news quoted the US President as saying to reporters outside the White House on Friday.
"He really wrote a beautiful three-page - I mean right from top to bottom - a really beautiful letter. And maybe I'll release the results of the letter, but it was very positive."
The missive was hand-delivered on Thursday, Trump said, quipping: "Don't have to worry about leaks."
This latest in a series of letters from Kim follows four rounds short-range rocket launches by North Korea in the space of two weeks.
Pyongyang says the launches are in response to this month's US-South Korean joint military exercises, a point that Kim emphasized in his message to Trump.
"He (Kim) wasn't happy with the tests, the war games. The war games on the other side with the United States," the president said. "And as you know, I've never liked it either. I don't like paying for it."
Washington suspended the twice-a-year drills after the historic June 2018 summit between Trump and Kim in Singapore and while the exercises were subsequently resumed, they are now conducted on a smaller scale.
Denounced by North Korea's communist government as "provocations," the drills formerly included a mock invasion of the North, complete with flyovers by US warplanes capable of delivering nuclear weapons.
Trump and Kim have met twice since the Singapore encounter: a summit in February in Hanoi; and a hastily arranged, largely symbolic event in late June on the border that divides the Korean peninsula.
On Friday, the president again downplayed the significance of the recent launches by North Korea.
"There have been no nuclear tests. The missile tests have all been short-range, no ballistic missile tests, no long-range missiles," Trump said.
The US Defense secretary said earlier Friday in Seoul that Washington was ready to resume talks with Pyongyang while maintaining sanctions on North Korea until it abandons its nuclear programme.
Mark Esper conveyed that message at the start of his meeting with his South Korean counterpart, Jeong Kyeong-doo.
Washington "will remain resolute in the enforcement of the UN Security Council resolutions until the North engages in the complete, verifiable, irreversible denuclearization of the Korean Peninsula," Esper said, according to South Korean news agency Yonhap.
( With inputs from IANS )