0914 GMT May 19, 2019
Trump told reporters traveling home to Washington with him aboard Air Force One from Argentina that "three sites" were in consideration for the meeting, a follow-up to their historic summit in Singapore in June, AFP reported.
"I think we're going to do one fairly (soon) – you know, into January, February, I think," said Trump, who had been in Buenos Aires for the G20 summit.
"We're getting along very well. We have a good relationship."
In the Argentine capital, Trump held separate bilateral talks with Chinese President Xi Jinping on Saturday that primarily focused on trade, but the US leader said Xi he had agreed to work with him "100 percent" on North Korea.
When asked Saturday if he would ever host the North Korean leader in the United States, Trump replied: "At some point, yeah."
In June, Trump and Kim opened up a face-to-face dialogue after months of trading military threats and pointed barbs.
The two leaders signed a vaguely worded document on denuclearization of the Korean Peninsula, but progress has since stalled as Washington and Pyongyang spar over the meaning of the document.
North Korea has taken few concrete steps to abandon its nuclear and ballistic missile programs.
US Secretary of State Mike Pompeo was due to meet with a top North Korean official in early November, but the meeting was abruptly put off, with Pyongyang insisting that Washington ease sanctions.
On Friday, Trump discussed the situation with South Korean President Moon Jae-in on the sidelines of the G20 summit.
The pair "reaffirmed their commitment to achieve the final, fully verified denuclearization" of North Korea, Trump's spokeswoman Sarah Sanders said.
They agreed on the need for "maintaining vigorous enforcement of existing sanctions to ensure the DPRK understands that denuclearization is the only path," Sanders said, using the North's official name, the Democratic People's Republic of Korea.
Moon's office welcomed Trump's comments about a second summit with Kim.
"We hope a concrete agenda and logistics will be determined soon," Yonhap News Agency quoted presidential spokesman Kim Eui-kyeom as saying.
But differences remain between Washington and Seoul on how to proceed with Kim, as the dovish Moon favors more robust engagement with the North.
North and South Korea have begun to remove landmines and destroy military bunkers at parts of their common border as part of efforts to improve long-strained relations.
They have also begun work to reconnect a train line and repair another rail link across the border.
Despite the warming ties, it remains unclear whether Kim will make his first-ever visit to the South this year, as Seoul is hoping.
Kim agreed to travel to Seoul after hosting Moon in Pyongyang in September for their third summit this year.
But prospects of a fourth Moon-Kim meeting have recently dimmed, with negotiations on denuclearizing the North grinding to a halt.