"A lot of progress" was being made, he said, speaking alongside South Korean President Moon Jae-in in New York.
"The relationship is very good and in some ways... extraordinary," he added.
Just a year ago, the US and North Korea were publicly speaking of a nuclear confrontation but Mr Trump had historic first talks with Mr Kim in June.
Speaking earlier, at UN headquarters in New York, he said that Secretary of State Mike Pompeo would arrange the next summit "in the immediate future".
No decision had yet been taken as to the venue, he said.
Mr Moon recently completed a high-profile, three-day visit to Pyongyang - the first by a Southern leader to the North's capital in a decade.