Speaking to reporters at the UN headquarters in New York, Mr Trump said Secretary of State Mike Pompeo would arrange it “in the immediate future”.
The president also said he saw “tremendous progress” in recent efforts to ease tension between the countries.
Just a year ago, the two nations were publicly speaking of a nuclear confrontation.
“That was a very dangerous time,” the president said. “This is one year later, a much different time.”
Mr Trump said Mr Kim had asked for a second summit through what he described as a “very warm” and “beautiful” personal letter.
This echoes comments made by South Korean President Moon Jae-in.
Mr Moon recently completed a high-profile, three-day visit to Pyongyang – the first by a Southern leader to the North’s capital in a decade.
On his return to Seoul, he said Kim Jong-un wanted “complete denuclearisation quickly”, and to focus on his country’s economic development.
“[Kim] said he hoped Mike Pompeo would visit North Korea soon, and also a second summit with Trump would take place in the near future, in order to move the denuclearisation process along quickly,” Mr Moon said.
Kim Jong-un met Mr Trump in Singapore in June, the first ever meeting between a sitting US president and a North Korean leader.
The pair signed a document pledging to establish new relations, work toward “complete denuclearisation” and to build a lasting peace on the peninsula.
Since then, while the North has made overtures to the South and tried to improve its image, little concrete change has occurred.
Some critics have raised concerns that the human rights situation in North Korea has been ignored to ensure better relations.