Moon is expected to arrive in New Zealand on Sunday for a three-day state visit after leaving Argentina, where he attended the Group of 20 summit.
The pair "reaffirmed their commitment to achieve the final, fully verified denuclearisation" of North Korea, Trump's spokeswoman Sarah Sanders said.
On Friday, Trump discussed the situation with South Korean President Moon Jae-in on the sidelines of the G20 summit.
Kim agreed to travel to Seoul after hosting Moon in Pyongyang in September for their third summit this year.
In his talks with Trump in Argentina, Moon earned some support for the Seoul summit from the USA leader - perhaps in a bid to entice Kim to make good on his pledge. "Of course, Kim's visit could take place a bit late ... but I think it doesn't matter whether it would happen before the end of this year or not".
Trump and Mr. Kim opened up dialogue on denuclearisation of the Korean peninsula after months of trading military threats and pointed barbs. It would be their second summit.
A USA think tank said last month it had identified at least 13 of an estimated 20 active, undeclared missile bases inside North Korea, underscoring the challenge for American negotiators hoping to persuade Kim to give up his weapons programs.
When asked about a future meeting, Trump said: "I think we're going to do one fairly (soon) - into January, February, I think".
"We're getting along very well".
In an interview with CNN on Saturday, Pompeo said the second US-North Korea summit would take place "shortly after the first of the year", without giving details. "We have a good relationship with Kim". Days after the cancellation, Moon held his second summit with Kim and helped him proceed with his meeting with Trump.
On November 7, the State Department announced that Secretary of State Mike Pompeo's meeting with Kim Yong Chol, vice chairman of the ruling Workers' Party of Korea Central Committee, which had been scheduled for November 8 in NY, would "take place at a later date".
"If the USA and North Korea continue to be left to break the stalemate by themselves, there will never be a compromise".
Differences over how to denuclearize the North have kept the 2 countries from deciding the date and venue of the next summit.
Moon's top press secretary, Yoon Young-chan, was quoted by Yonhap news agency saying that Trump "asked that South Korea and the United States closely cooperate so the next summit will be another historical milestone in the process of denuclearizing the Korean Peninsula".