Trump, who is holding his second summit with Kim following their historic meeting in Singapore last June, was referring to a timeline on denuclearization and the dismantlement of North Korean weapons production facilities.
President Trump urged Kim Jong-un to give up his nuclear weapons and pursue Vietnam's path to prosperity by embracing capitalism and foreign investment as he began his second summit with the North Korean leader in Hanoi.
White House press secretary Sarah Sanders said the leaders had a "very good and constructive meeting" and discussed ways to advance "denuclearization and economic driven concepts".
"I'd much rather do it right than do it fast", Trump said. "Because North Korea spends so much money on nuclear development, and because of that nuclear development, North Korean became one of the superpowers of the world".
For his part, Kim came to Hanoi hoping for substantial relief from sanctions.
Skeptics say such agreements would leave in place a significant portion of North Korea's nuclear-tipped missiles while robbing the United States of its negotiating leverage going forward. These uncertainties are just the reason why we need to think more seriously about the issue of securing our security self-determination.
U.S. President Donald Trump said on Thursday he had walked away from a nuclear deal at a summit with North Korean leader Kim Jong Un because of unacceptable North Korean demands to lift punishing U.S. -led sanctions.
The US President told reporters he thought the talks would be very successful - but warned he was "not walking back on denuclearisation".
"Chairman Kim and I had a great talk about that last night", Mr Trump added.
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Earlier Kim and Trump, seated across from each other at a conference table, appeared confident of progress.
Trump blamed the breakdown on North Korea's insistence that all the punishing sanctions the USA has imposed on Pyongyang be lifted without the North committing to eliminate its nuclear arsenal.
But the U.S. leader said Thursday he was in "no rush", appearing to back away from any expectations the much-anticipated meeting could produce any major breakthroughs.
North Korea has taken steps to dismantle a rocket engine testing site and blow up a tunnel at a nuclear testing site.
Moon had hoped to follow the Trump-Kim meeting with his own fourth summit with Kim, preferably in South Korea's capital, Seoul.
"He said he's not going to do testing of rockets or missiles or anything that has to do with nuclear", the U.S. president said.
The U.S. side said "that wouldn't work", he said, adding that North Korea was unwilling to completely freeze its weapons programs.
While North Korea's official media said Kim and Trump had made a decision to continue talks, its Vice Foreign Minister Choe Son Hui has struck a more negative tone, telling reporters Kim "might lose his willingness to pursue a deal".
"Yongbyon is the heart of North Korea's nuclear program", nuclear scientist Dr. Siegfried Hecker told NBC.
USA concessions could include opening liaison offices or clearing the way for inter-Korean projects, but critics say Trump risks squandering vital leverage if he gives away too much, too quickly.