"If it would be appropriate for me to meet with him, I would - absolutely. I would be honoured to do it," he told news organisation Bloomberg on Monday.
The previous day he described Mr Kim as a "pretty smart cookie".
The comments come amid escalating tensions over North Korea's nuclear programme.
The White House issued a statement following Mr Trump's remarks, saying North Korea would need to meet many conditions before any meeting between the two leaders could take place.
Spokesman Sean Spicer said Washington wanted to see the North end its provocative behaviour immediately.
"Clearly conditions are not there right now," he added.