"We have a good relationship", Trump told reporters aboard Air Force One on his return from a G20 summit in Argentina.
Trump was also quoted as saying that at some point, he would invite Kim to the United States.
Mr Trump has frequently described a warm personal relationship with Mr Kim, arguing that this rapport would help him succeed at a diplomatic breakthrough that has eluded USA presidents since the 1950s. But both Moon and Trump say fears of war on the Korean Peninsula have eased sharply.
However, Washington is taking a more cautious approach towards Pyongyang amid sparring over interpretation of a vaguely worded denuclearisation document signed at the Singapore summit.
While the announcement establishes a window for a summit between Trump and Kim, it now appears that the two leaders are more likely to meet toward the later end of early 2019. North Korea has taken few concrete steps to abandon its nuclear and ballistic missile programs.
Former UN Secretary-General Ban Ki-moon urged North Korean leader Kim Jong Un on Monday to take concrete steps toward complete denuclearisation and gain the trust of the global community if he wants sanctions lifted. The pair "reaffirmed their commitment to achieving the final, fully verified denuclearisation" of North Korea, Trump's spokeswoman Sarah Sanders said. But differences have emerged between Washington and Seoul on how to proceed with Kim, as the dovish Moon has long favoured engagement with the North. "It is North Korea which should clearly state and show by taking actions, then I am sure that there will be no reason why the Security Council of the United Nations will continue to impose the sanction on North Korea", Ban said. They have also begun work to reconnect a train line and fix another rail link across the border.
"Ever since South and North Korea were divided into two countries, this will be the first time that a North Korean leader will visit South Korea", Moon said through an interpreter.
Speaking at a news conference following that announcement, President Moon said Kim would fly to Seoul within the year, unless "special circumstances" arose.
A visit by North Korean leader Kim Jong Un to Seoul before the end of the year remains possible, South Korean President Moon Jae-in reiterated to press on Tuesday.