Citing the possibility of meeting Kim at the DMZ, that separates the two Koreas, Trump told the media here: "They want to do it. I want to do it."
"I know that it's very difficult in terms of logistics, security, very complicated."
His remarks came before a summit with his South Korean counterpart Moon Jae-in at the Presidential Palace here, reports Yonhap News Agency.
Trump also emphasized his good personal relations with Moon.
"I think I can say with great confidence that the relationship with South Korea has never been stronger or better," Trump said. "Good chemistry between the leaders."
Trump earlier invited Kim to a "handshake" meeting at the DMZ, apparently at the truce village of Panmunjom, or the Joint Security Area, as the venue.
With the Twitter invitation for Kim to shake hands with Trump at the DMZ, Trump gave the whole world "big hope", Moon said.
Moon agreed that such a DMZ session, if realized, would be a "very historic" event.
"It will set up a very big landmark in the denuclearization of the Korean Peninsula and the Korean Peninsula peace process."
Moon said he would accompany Trump on his DMZ visit but suggested that he won't meet Kim himself.
Meanwhile, Koryo Tours, a Beijing-based tour agency specializing in North Korea tour programmes, said on its Twitter account: "The DMZ is closed to tourists visiting from the North Korean side today," Yonhap reported.
Following the summit, Trump and Moon will have an expanded working-lunch session to include more officials of the two sides that will be followed by a joint press conference.
It's their first one-on-one summit since the April one at the White House.
Trump arrived at the Presidential Palace after attending a "business leaders' event" with the heads of some South Korean conglomerates.
During his previous visit to South Korea in November 2017, Trump pushed for a DMZ trip with Moon by chopper. But it was canceled due to bad weather.
( With inputs from IANS )