Trump said it is even possible that a meeting could take place on June 12 in Singapore as originally planned, although that appeared unlikely. The optimism threw a new twist into an already chaotic run-up to what would be the most significant foreign policy gambit of Trump's presidency.
"We'll see what happens. We are talking to them now," Trump told reporters as he left the White House on Friday morning. "They very much want to do it. We'd like to do it."
"It could even be the 12th," he added.
Just 24 hours earlier the White House had released a letter from Trump to Kim in which Trump blamed the cancellation on "tremendous anger and hostility" coming from North Korea and said that the rogue nation had "lost a great opportunity for lasting peace."
A senior administration official told reporters Thursday that North Korea had stood up U.S. officials for a planning meeting this month, and had balked at finalizing logistics. There was no longer enough time to make the summit happen on June 12 even if North Korea suddenly changed tack, the official said.
That calculation appeared to have changed Friday, with Trump's upbeat remarks and word that a logistical advance team might still travel to Singapore on Sunday as originally planned.
"There are still a few hours left to decide," White House press secretary Sarah Huckabee Sanders said. "If it takes place on June 12th, we'll be ready. If it takes place on July 12th, we'll be ready."
Trump heightened the prospects of getting the talks back on track on Friday night, tweeting that the United States was having "very productive talks with North Korea about reinstating the Summit."
Trump said if the summit does take place, it "will likely remain in Singapore on the same date, June 12th."
Trump administration officials said the summit could be salvaged if North Korea offers an explicit commitment to a rapid denuclearization process - an assurance they had refrained from making in recent days.
"We were not receiving the right signals previously, so hopefully that will change as we move forward," Robert Palladino, a National Security Council spokesman, said Friday. "It's important that North Korea commit to the quick denuclearization of the Korean Peninsula. When North Korea is prepared to act in good faith, we will be ready."
Harry Kazianis, an Asia expert at the Center for the National Interest, said U.S. officials have been frustrated in recent days by vague aspirational statements about denuclearization that offered few assurances about timing or substance.
"U.S. officials are asking the North Koreans one basic question: Are you serious about denuclearization or not?" He said. "If the answer is yes, but only in an aspirational sense, the summit is not happening. If the answer is yes, and here's what it would like, then they'll have a summit."
Trump's abrupt cancellation was partly a response to concern that North Korea might scrap the meeting first, U.S. officials said, and fulfilled Trump's pledge to "walk away" from talks if he felt they were unproductive.
But hours later, North Korea responded with a surprisingly conciliatory statement that included praise for Trump.
"We have inwardly highly appreciated President Trump for having made the bold decision, which any other U.S. president dared not, and made effort for such a crucial event," as the June 12 summit, Kim Kye Gwan, North Korea's vice foreign minister, said in a statement on state-run media.
"We would like to make known to the U.S. side once again that we have the intent to sit with the U.S. side to solve problem regardless of ways at any time," the vice foreign minister said in the statement that appeared Thursday evening in the United States.
The restrained response from Kim, a longtime nuclear negotiator, was notable in part because his sharp criticism of Trump national security adviser John Bolton earlier this month had been the first public sign of trouble in the summit planning.
It was not clear whether North Korea had been shaken by Trump's cancellation and was trying to resurrect the June 12 meeting, or merely sought to regain diplomatic footing by appearing open to negotiations at some point.
Analysts are divided about whether North Korea has made a strategic calculation that negotiations are the best way to get out from under stringent economic sanctions and preserve the Kim dynasty. Few think Kim Jong Un is willing to entirely forfeit his nuclear weapons arsenal in any deal with the United States or other world powers.
In a Friday morning tweet, the president characterized the new statement as "warm and productive" and "very good news."
Trump has pointed to his willingness to sit down with Kim, and to walk away if necessary, as evidence of his negotiating prowess.
Asked by a reporter Friday whether North Korea had been playing games, Trump said: "Everybody plays games. You know that better than anybody."
Trump made his comments as he headed to Annapolis, Maryland, to speak at the U.S. Naval Academy commencement ceremony. On Twitter before his departure, he also accused Democrats of "rooting against" his administration's efforts.
House Minority Whip Steny Hoyer, D-Md., on Thursday called Trump's letter to Kim relaying his decision "another demonstration of President Trump's treating of critical negotiations as if they were just another real estate deal."
State Department spokeswoman Heather Nauert on Friday said Trump administration officials were not surprised by the course of events and remain hopeful that the meeting between Trump and Kim will happen at some point.
"We never expected it to be easy, so none of this comes as a surprise to us," she said.
Nauert said that Secretary of State Mike Pompeo, White House national security adviser John Bolton and others remain in close coordination about a path forward.
"So we hope that the meeting will go forward at some point," Nauert said. "We weren't getting the right signals previously, so hopefully we will in the future. But we didn't want to go to a meeting just for the sake of going to a meeting. There had to be something to come out of it."
Pompeo spoke Thursday with his South Korean counterpart, Nauert said in a statement later Friday, and the two "committed to remaining closely coordinated in all of their efforts to create conditions for dialogue with North Korea and agreed that must continue until North Korea embraces denuclearization."
South Korean President Moon Jae-in had appeared blindsided by Trump's cancellation notice Thursday.
Secretary of Defense Jim Mattis echoed Trump's optimism Friday while speaking to reporters before a meeting with Danish officials at the Pentagon.
"We have got some, possibly some good news on the Korea summit, where it may, if our diplomats can pull it off, may have it back on even," Mattis said. "Our president just sent out a note about that a few moments ago. That is a usual give and take, you know, of trying to put together big summits and stuff. The diplomats are still at work."
Mattis was also asked whether the U.S. military would increase military exercises after the summit cancellation.
"We are not changing anything right now, it is steady as she goes," he said. "The diplomats are in the lead and in charge, and we give them our best wishes to have a fruitful way forward."
Trump made no mention of North Korea during his address Friday to the graduating class of the Naval Academy, though he spoke of the importance of a strong military to ensure peace.
"We are respected again," Trump said. "I can tell you that. We are respected again."
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The Washington Post's Missy Ryan contributed to this report.
(Except for the headline, this story has not been edited by NDTV staff and is published from a syndicated feed.)