Though Trump's top aides emphasized that the administration is examining all diplomatic measures to rein in Pyongyang, they made clear that military options remain on the table.
"If North Korea keeps on with this reckless behavior, if the United States has to defend itself or defend its allies in any way, North Korea will be destroyed," Nikki Haley, the U.S. ambassador to the United Nations, said Sunday on CNN's "State of the Union." "None of us want that. None of us want war. But we also have to look at the fact that you are dealing with someone [in Kim] who is being reckless, irresponsible and is continuing to give threats not only to the United States, but to all of its allies. So something is going to have to be done."
The question remains, however, how realistic the Trump administration's threats are as the North quickly advances its nuclear and ballistic missile capabilities. Trump's latest tweets came two weeks after North Korea tested a nuclear device that experts said measured at 250 kilotons, 17 times the force of the atomic bomb that destroyed Hiroshima in World War II.
Trump warned Kim last month that the North would feel the "fire and fury" of the United States if the regime continued its threats and destabilized the Korean Peninsula and East Asia. But Kim promptly responded with new threats and a round of new weapons tests.
Trump is scheduled to join Moon and Japanese Prime Minister Shinzo Abe at a working lunch Thursday in New York, on the sidelines of the U.N. meetings, to discuss North Korea, White House aides said. Yet Trump will not have the opportunity to meet with Xi Jinping of China and Vladimir Putin of Russia; both leaders are skipping the annual gathering.
Last week, Haley touted the U.N. sanctions on the North, saying that, if enacted, they would cut off 30 percent of oil imports and curtail 90 percent of Kim's exports, putting a major economic pinch on a government that has long struggled to provide for the nation's estimated 25 million people.
Yet Trump said last week that he and Secretary of State Rex Tillerson are skeptical that the sanctions will have a significant impact on North Korea's nuclear ambitions. Administration officials reaffirmed the United States' long-standing policy that the North must agree to relinquish its nuclear arsenal as a prerequisite for direct diplomatic talks.
"He's going to have to give up his nuclear weapons, because the president has said that he is not going to tolerate this regime threatening the United States and our citizens with a nuclear weapon," national security adviser H.R. McMaster said on ABC's "This Week."
Trump, McMaster added, has "been very clear about that, that all options are on the table."
At the same time, the administration signaled that it is not pursuing regime change, a position that could help persuade Beijing to play a stronger role in pressuring Kim. China facilitates about 90 percent of North Korea's trade and provides its oil.
Yet Tillerson said North Korea does not appear to be interested in denuclearization talks.
"I'm waiting for the regime in North Korea to give us some indication that they're prepared to have constructive, productive talks," he said on CBS's "Face the Nation."
"We have tried a couple of times to signal to them that we're ready when they're ready," Tillerson added, "and they responded with more missile launches and a nuclear test. All they need to do to let us know they're ready to talk is to just stop these tests, stop these provocative actions, and let's lower the threat level and the rhetoric."
Many U.S. allies in Europe and elsewhere are strongly opposed to any use of force that could further destabilize the Korean Peninsula and East Asia. The two unanimous U.N. Security Council votes for sanctions in recent weeks have marked a new level of alarm from those allies, as well as Moscow and Beijing.
But McMaster said Washington isn't assuming the sanctions will work or buy time.
"We all have our doubts about whether or not that's going to be enough," he said, "and so we have to prepare all options. We have to make sure all options are under development to ensure that this regime cannot threaten the world with a nuclear weapon."
Analysts have said the North has shown rapid improvements in its ballistic missile and nuclear technologies. Recent missile tests have demonstrated the range to potentially strike the continental United States, along with the technical capacity to mount a miniature nuclear device on a missile, analysts said.
That has accelerated the urgency in Washington, at the White House and on Capitol Hill. Asked on CNN whether the Trump administration should continue to deny the North diplomatic talks until it ends its nuclear program, Sen. Dianne Feinstein (D-Calif.), a member of the Senate Intelligence Committee, said no.
"I think that North Korea is not going to give up its program with nothing on the table," she said. "I think that what could happen is that we could have reliable verification of a freeze of both the nuclear program and the missile arsenal, and that we could conceivably talk China into supporting that kind of a freeze, because it would carry with it no regime change and no war."
Carol Morello in Washington contributed to this report.