These developments, combined with a surprise meeting Saturday between Kim and South Korea's president, Moon Jae-in, rekindled hopes of progress toward halting the North's nuclear weapons program.
The two leaders held their meeting on the Northern side of the truce village of Panmunjom, inside the DMZ.
After cancelling the scheduled summit last Thursday, blaming North Korea's "tremendous anger and open hostility", Mr Trump tweeted on Friday that "very productive talks" had taken place with the North on reinstating the summit with Mr Kim.
Just a day after the cancellation, however, Trump said the United States and North Korea were "having very productive talks" and there was still a chance the meeting would be held in Singapore on the same date.
Following Trump's cancellation of the summit, North Korea issued a statement declaring it was still open to talks.
Trump, speaking Friday to graduates at the U.S.
The North has said it will refuse to participate in talks where it would be unilaterally pressured to give up its nukes. "You talk about your nuclear capabilities, but ours are so massive and powerful that I pray to God they will never have to be used", Trump wrote.
At the same time, it also was increasingly clear that the administration's goal of "total denuclearization" of the Korean peninsula was interpreted differently in North Korea, which analysts say is unlikely to give up all of its nuclear capabilities and especially not in short order, as the US has demanded.
Koh said Moon would try to persuade Kim to accept an alternative approach advocated by Seoul, in which the North's comprehensive commitment and credible actions toward denuclearization are followed by a phased but compressed process of declaration, inspection, and verifiable dismantling.
Kim Kye Gwan said North Korea appreciated Trump for having made the bold decision to work toward a summit.
North Korea also went ahead with a plan to destroy its only known nuclear site on Thursday, the most concrete action yet since pledging to cease all nuclear and long-range missile tests last month.
The White House was unhappy about what it considered to be a "trail of broken promises" by North Korea - including failure to show up for summit preparatory talks and complaints about the latest US-South Korean joint military exercise. "We're going to see what happens", he said, according to White House media pool reports. There, former leader Colonel Gaddafi gave up his nascent nuclear programme only for him to be killed by Western-backed rebels a few years later.
North Korea issued a statement on Friday saying it was still "willing to give the USA time and opportunities" to reconsider talks "at any time, at any format".
The conversation appeared to amount to an effort to reassure a key USA ally that Trump did not intend to further escalate tensions with North Korea and that Washington still viewed diplomacy as its primary tool for dealing with Pyongyang.
Saturday's meeting is only the fourth time serving leaders of the two Koreas, who remain technically at war, have ever met.
Japanese Prime Minister Shinzo Abe, at the same forum, regretted the cancellation of the summit and said the world should keep doing its bit to make it happen.
However, the United States president later tweeted that the summit could be back on and held in Singapore as originally planned - with Seoul expressing relief that the prospect of talks had been revived. But North Korean state media, in a measured response, stated that they were still open to meeting the USA president. Photos showed that Kim's sister, Yo-jong, widely believed to be a key player in her brother's inner circle, was also present.
The Friday morning (local time) tweet came a day after he withdrew from the June 12 summit.