The meeting "could happen soon," Trump said on the sidelines of meetings at the 74th United Nations General Assembly in New York City.
"We’ll see," he said when asked what it would take for another face-to-face with Kim. "Right now, people would like to see that happen. I want to know what’s going to be coming out of it. We can know a lot before the summit takes place."
Denuclearization talks have been stalled for several months. While North Korea again embarked on so-called missile diplomacy -- conducting eight weapons tests since July -- Pyongyang said earlier this month the country was open to talks with Trump in "late September," according to state-run media.
North Korea has demanded that the U.S. lift crippling economic sanctions in order to move forward and possibly dismantle its nuclear program on the Korean Peninsula.
South Korean President Moon Jae-in, who met with Trump on Monday, said he expects the nations to hold another summit, and for talks to resume.
“I always marvel at your imagination and bold decision-making,” Moon said to Trump.
The White House reportedly received a letter last week from North Korea inviting Trump to visit Pyongyang.
Trump has previously touted his good relationship with Kim Jong Un, saying in August he received a "beautiful letter" from the dictator.
"The relationship has been very good,” Trump reiterated Monday. “We’ll see what happens. But we’d like to see if we can do something. And if we can, that would be great. And if we can’t, that’s fine."
In June, Trump became the first sitting U.S. president to step foot in in the Hermit Kingdom, meeting briefly with Kim at the Demilitarized Zone (DMZ) that divides North and South Korea.
It was the first time the two had met in person since their failed second summit in Hanoi, Vietnam, in February.
Trump and Kim's historic first summit took place in June 2018 in Singapore.
The Associated Press contributed to this report.