Trump said on Tuesday at the White House that the meeting would "probably" not be in Singapore, where their historic first talks took place in June.
The pair discussed ending the reclusive state's nuclear weapons program and hostilities between Washington and Pyongyang.
Trump said that in terms of timing, the summit "won't be too far away," later telling reporters it would take place after the November 6 midterm elections.
He also said that there could "eventually" be a meeting on US soil.
"On their soil also," Trump added.
US Secretary of State Mike Pompeo met with Kim on Monday in Pyongyang to discuss the next summit.
"I returned late last night from North Korea from a trip where we made real progress. While there is still a long way to go and much work to do, we can now see a path to where we'll achieve the ultimate goal which is the full and final verified denuclearization of North Korea," Pompeo told reporters at the White House.
"We will, in short order, be able to talk about when the president will get to meet with him at what will be the second summit." Trump talked of "incredible progress," hailing the absence of missile or nuclear tests this year and the recent return of remains of US service members killed during the Korean War.
"I like him, he likes me, the relationship is good."
(With inputs from agencies.)