Asked about Trump's description of Kim, national security adviser H.R. McMaster said "Rocket Man" was "a new one and I think maybe for the president".
Trump tweeted Sunday that long lines for gas are forming in North Korea and called it "too bad".
However, John DeLury of Seoul's Yonsei University, a leading North Korea expert, believes Kim's tough talk might signal an "opening" to negotiations with the U.S.as he builds his nuclear arsenal - talks that may be the only answer to the situation in Pyongyang. Trump's United Nations ambassador, Nikki Haley, said Sunday on CNN's State of the Union that North Korea was starting to "feel the pinch" from the additional sanctions of more than $1 billion.
"We have pretty much exhausted all the things that we can do at the Security Council at this point", ambassador Nikki Haley told CNN.
But Haley warned of a tougher US response to future North Korean provocations, and she said she would be happy to turn the matter over to Defense Secretary James Mattis "because he has plenty of military options".
Speaking at a White House news conference earlier this week, the US President said he would prefer not to use military action against the nuclear state, but warned "nothing is inevitable". North Korea also recently tested what it said was a hydrogen bomb.
"And none of us want that". Pyongyang's military has conducted three ballistic missile tests since US President Donald Trump told reporters the country would be met with "fire and fury and, frankly, power, the likes of which the world has never seen before" if it kept up threats to the US and its allies.
North Korea has launched a trio of missiles in recent weeks and tested a bomb that was its most powerful to date.More news: Arsenal vs Cologne — UEFA Europa League
North Korea said on Saturday it aimed to reach an "equilibrium" of military force with the United States. Haley said that wasn't an empty threat from the president but, when asked, she declined to describe the president's intentions. "None of us want war".
McMaster said Trump's scheduled speech Tuesday at the U.S. General Assembly in NY will "communicate his vision for America's role in the world" in three areas: protecting American people, promoting American prosperity and promoting accountability and sovereignty.
"What we are seeing is they continue to be provocative, they continue to be reckless".
Secretary of State Rex Tillerson said he was waiting for the North to express interest in "constructive, productive talks".
The United States has pushed to cut off oil and gas to the North, but it settled last week for modest cutbacks under a U.N. resolution.
China has urged the United States to refrain from making threats to North Korea. "And they have responded with more missile launches and a nuclear test".
The Wall Street Journal and other news organizations reported Saturday that Trump administration officials, speaking at a climate meeting in Montreal, discussed a compromise that would involve staying in the deal but revising USA climate change goals.
Still, they were less harsh than those initially proposed by the USA - and Nikki Haley, the US ambassador to the United Nations, spoke ominously of the possible failures of diplomacy.