"This is a real threat to the world, whether we want to talk about it or not," Trump said Monday at a lunch for ambassadors of countries who sit on the Security Council.
"North Korea's a big world problem, and it's a problem we have to finally solve. People put blindfolds on for decades and now it's time to solve the problem."
Trump's comments come as the USS Michigan, one of US's most powerful submarines, arrived in South Korea
in an apparent show of US force and senators have also been summoned to the White House to be briefed Wednesday by top officials about the threat posed by North Korea.
The US president later told a reception of conservative journalists that Kim wasn't the strong leader he likes to portray himself as.
"I'm not so sure he's so strong like he says he is, I'm not so sure at all," Trump told the reception in comments confirmed to CNN by the White House.
The flurry of activity comes as North Korea marks the 85th anniversary of the founding of its army Tuesday -- a significant date in the country's calendar, which it used to conduct a "large-scale" live-fire military drill.
Is Trump taunting Kim?
Commentators said that Trump, with his strong rhetoric, was backing himself into a corner over North Korea, leaving him in a potentially dangerous position.
"It's been all about pressure and brinksmanship and military tools and tactics and chess something, and now to literally, directly taunt the leader of North Korea is like poking the hornet's nest yet again," said Paul Carroll, program director at the Ploughshares Fund, a group working to eliminate nuclear weapons.
"If you do this kind of ratcheting (up) of rhetorical tensions but also sending military hardware into the region, if you do that absent of any kind of offramp or door or window for the North Koreans to consider walking through and reengaging in any sort of discussions, bad things are going to happen," he added.
Tong Zhao, a fellow at the Carnegie Tsinghua Center for Global Policy in Beijing, said singling out the North Korean leader in this way could provoke him.
"Being called a weak leader (by a US president), I think, this is the first time I remember hearing it. It could potentially provoke North Korea ... Being called weak will only encourage them to appear more strong," he told CNN.
Increased US presence
Trump's statements came as
the USS Michigan arrived at a South Korean port Tuesday
, in the city of Busan.
A guided-missile submarine, it can carry up to 154 Tomahawk cruise missiles and is able to stealthily deploy up to 66 Navy SEALs or other special operations troops.
"I think that's a real message because the submarine carries 150 something cruise missiles, and it's a very powerful striking force," Zhao said.
"It's hard to predict how North Korea will respond."
The USS Carl Vinson, is heading towards South Korea as part of a long-promised deployment by the Trump administration
and South Korean and US Navy destroyers began joint naval drills in the Yellow Sea Tuesday.
Rare full US Senate briefing planned
Amid the military manoeuvrings, the White House has taken the unusual step of calling the entire US Senate in for a briefing on North Korea Wednesday afternoon.
"The stakes are much higher -- the US believes that it is facing a critical juncture where North Korea is about to obtain a nuclear capability against the US homeland," Zhao said.
Only senators will be allowed to attend, where they will be briefed by Secretary of State Rex Tillerson, Defense Secretary James Mattis and Joint Chiefs Chairman Gen. Joseph Dunford.
Diplomatic talks between the United States and East Asian powers on North Korea also intensified this week.
On top of trilateral talks in Tokyo between Japan, South Korea and the US taking place Tuesday, US President Donald Trump and Chinese President Xi Jinping spoke over the phone Monday night, the second time in as many weeks.