In an interview with Reuters, Trump, who held a landmark summit with Kim on June 12, said he believed North Korea had taken specific steps toward denuclearization, despite widespread doubts about Kim's willingness to abandon his arsenal.
While insisting that "a lot of good things are happening" with North Korea, Trump complained that China was not helping as much as it had in the past because of its trade dispute with the United States.
Trump, who faced the North Korean challenge as soon as he took office in January 2017, said he had only been working on the North Korean issue for three months whereas his predecessors had been working on it for 30 years.
"I stopped (North Korea's) nuclear testing. I stopped (North Korea's) missile testing. Japan is thrilled. What's going to happen? Who knows? We're going to see," he said.
At their summit in Singapore, Kim agreed in broad terms to work toward denuclearization of the Korean peninsula but North Korea has given no indication it is willing to give up its weapons unilaterally as the Trump administration has demanded.
Trump has hailed the Singapore summit as a success and went as far as saying North Korea no longer posed a nuclear threat.
In the interview, Trump credited his "great chemistry" with Kim for easing a nuclear standoff that last year raised fears of a new Korean war.
"I like him. He likes me," he said. "There's no ballistic missiles going up, there's a lot of silence ... I have very good personal relations with Chairman Kim, and I think that's what holds it together."
Asked whether another meeting with Kim was on the horizon, Trump said: "It's most likely we will, but I just don't want to comment." But he offered no details on the timing or venue.
Critics say Trump made too many concessions to Kim by agreeing to hold the summit in the first place and then suspending joint U.S.-South Korea military drills while gaining little in return.
Asked whether North Korea had taken specific steps to denuclearize other than blowing up its main nuclear bomb test site ahead of the summit, Trump: "I do believe they have." But he did not elaborate.
He also pointed to North Korea's pre-summit handover of three American detainees.