Tensions on the Korean peninsula have been running high for weeks, with signs the North might be preparing a long-range missile launch or a new nuclear test -- prompting tough talk from Washington, which has refused to rule out a military strike in response.
But the Trump administration is also spearheading a renewed diplomatic push to bring Pyongyang back to the negotiating table.
Trump has so far placed his biggest bet on getting China to use its leverage to pressure Pyongyang to change its behavior -- a strategy that has failed to produce results in the past.
The Republican president has also said he is ready to act alone in the stand-off, however -- and on Monday signaled that this could involve face-to-face talks with Kim, who has yet to meet a foreign leader since taking power.
"If it would be appropriate for me to meet with him I would, absolutely. I would be honoured to do it," Trump said in an interview with Bloomberg.
"If it's under the, again, under the right circumstances. But I would do that," Trump said.
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In the latest rhetoric to fuel jitters across the region, North Korea warned Monday that it was prepared to carry out a nuclear test "at any time and at any location" set by its leadership.
The regime will continue bolstering its "preemptive nuclear attack" capabilities unless Washington scraps its hostile policies, a spokesman for the North's foreign ministry said in a statement carried by the state-run KCNA news agency.
"The DPRK's measures for bolstering the nuclear force to the maximum will be taken in a consecutive and successive way at any moment and any place decided by its supreme leadership," the spokesman added, apparently referring to a sixth nuclear test and using the North's official name, the Democratic Republic of Korea.
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The North has carried out five nuclear tests in the last 11 years and is widely believed to be making progress toward its dream of building a missile capable of delivering a warhead to the continental United States.
It raises the tone of its warnings every spring, when Washington and Seoul carry out joint exercises it condemns as rehearsals for invasion. But this time fears of conflict have been fueled by a cycle of threats from both sides.
The joint drills have just ended, but naval exercises are continuing in the Sea of Japan (East Sea) with a US strike group led by the aircraft carrier USS Carl Vinson.
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North Korea's Kim is 'a pretty smart cookie,' Trump says
CIA director Mike Pompeo was in South Korea on Monday, the US embassy in Seoul confirmed, following reports of an unannounced visit as tensions mount on the peninsula.
Pompeo's visit coincided with news that the controversial US missile defense system known as THAAD -- whose deployment has angered China -- is now operational in South Korea.
"It has reached initial intercept capability," a US defense official told AFP on condition of anonymity.
'A pretty smart cookie'
Seoul regularly warns that Pyongyang can carry out a test whenever it decides to do so.
Pyongyang's latest attempted show of force was a failed missile test on Saturday that came just hours after US Secretary of State Rex Tillerson pressed the UN Security Council to raise pressure on the North.
Trump on Sunday repeated his determination to resolve the threat posed by North Korea, warning in a CBS interview: "We cannot let what's been going on for a long period of years continue."
But the US leader also offered some backhanded praise for Kim, saying he had faced a formidable challenge in taking over the country at a reported age of 27 after his father's death in 2011.
"He's dealing with obviously very tough people, in particular the generals and others. And at a very young age, he was able to assume power," Trump said on CBS's "Face the Nation."
"So obviously, he's a pretty smart cookie," he said.
