Trump, who is visiting Osaka for 14th G-20 Summit, had sent a personal letter the North Korea leader earlier this week.
"While there, if Chairman Kim of North Korea sees this, I would meet him at the Border/DMZ just to shake his hand and say Hello (?)!" he said.
In April, North Korean leader Kim said that he will wait for the U.S.to make a "courageous" decision until the end of this year, apparently urging Washington to ease what it sees as unilateral denuclearization demands.
When a reporter asked him about whether he has "heard anything" from North Korea, he said, "We have".
But the president says he will now allow USA companies to sell components to Huawei again.
Trump's summit with Kim in Vietnam earlier this year collapsed without an agreement to denuclearize the Korean Peninsula. Whether it happens or not remains an open question, one even Trump claimed to be unable to answer after he raised the prospect in a tweet.
There had been talks of a third summit between President Trump and North Korea's Kim Jong-un.
Trump made the offer in a morning tweet ahead of a second day of meetings at the Group of 20 summit here. Pyongyang has yet to respond to Moon's offer for dialogue. The most likely locations for visits were Osan Air Base and Camp Humphreys, considering they are larger than other US bases in South Korea.
North Korea is heavily dependent on China in the import of essentials, and the two countries have a long history of partnership in matters related to defence and other critical bilateral relations.
After returning to South Korea, Moon greeted Trump, who arrived at an American airbase in Osan for a dinner. "They agreed that it is an important time for progress in the peaceful approach towards North Korea and substantive progress is needed in the resumption of US-North Korea talks". Pak is an iconic figure in South Korea's golf circles, and EXO met with Ivanka during the closing ceremony of the 2018 PyeongChang Winter Olympics.
It was unclear if this meant a shift in the White House's stance, as it has insisted on a maximalist outcome - "the final, fully verified denuclearization of North Korea" first and foremost. But Trump says he thinks she "was given far too much credit for what she did" and says her answer came "right out of a box". As he and Xi declared a cease-fire in their countries' trade war, Trump enthused that "Chinese culture is an incredible culture", and repeatedly called Xi his friend.
Despite the deadlocked nuclear negotiations, both Trump and Kim have described their personal relationship as good.
The Joint Security Area (JSA) has always been mentioned as a possible venue for a Trump-Kim meeting.