"That's why I call him Rocket Man," Trump told reporters during a meeting with the head of NATO in London He said he hoped Kim would denuclearise, but added: "we'll find out."
North Korea fired two short-range projectiles into the sea off its eastern coast last week in the latest test of its large multiple-rocket launcher. It was seen as an effort to remind the United States of a year-end deadline Kim has set for Washington to show flexibility in stalled denuclearisation talks.
The United Nations Security Council is due to meet behind closed doors on Wednesday - at the request of France, Britain and Germany - to discuss the latest missile launches by Pyongyang, diplomats said. The 15-member Security Council banned North Korea's use of ballistic missiles in 2006.
North Korea earlier on Tuesday accused the United States of trying to drag out denuclearisation talks ahead of the U.S. presidential election next year.
Trump said he was also pressing ahead with negotiations with allies South Korea and Japan to shoulder more of the cost of stationing U.S. troops in those countries.
He said South Korea last year agreed to pay nearly $500 million a year more for U.S. "protection," and added the United States now wanted additional commitments.
Asked if it was in the U.S. national security interest to have U.S. forces stationed on the Korean peninsula, Trump said: "It can be debated. I can go either way. I can make arguments both ways."
"But I do think this, I think if we're going to do it, they should burden-share more fairly," Trump said.