Trump and Kim are due to meet in Hanoi next week to discuss progress on the issue of North Korea's nuclear program since they first met in Singapore last year.
"Chairman Kim and I have a very good relationship. I wouldn't be surprised to see something work out," Trump said, without providing further details.
Trump reiterated his view that North Korea has "great" potential for economic development and suggested the Vietnam summit would not be the last.
At their landmark meeting in Singapore last year, the US and North Korean leaders produced a vaguely worded document in which Kim pledged to work toward "the denuclearization of the Korean peninsula."
But progress has since stalled, with the two sides disagreeing over what that means.
Trump on Tuesday said that while he hoped his next meeting with Kim would yield much progress, he was also in "no rush" for results.
The two men are set to meet in Hanoi on February 27-28.
White House spokeswoman Sarah Sanders meanwhile said Trump spoke by phone with Japan's Prime Minister Shinzo Abe and the two men "reaffirmed their commitment to achieving the final, fully verified denuclearization of" North Korea.
'The sanctions are on in full. I haven't taken sanctions off,' Trump told reporters at the White House.