Trump, who has urged China to do more to rein in its impoverished ally and neighbor, said in a Tweet that North Korea was "looking for trouble" and the United States would "solve the problem" with or without Beijing's help.
Tension has escalated sharply on the Korean peninsula amid concerns that reclusive North Korea may soon conduct a sixth nuclear test and after Washington said at the weekend it was diverting the aircraft carrier strike group Carl Vinson toward the Korean peninsula in a show of force.
"We are sending an Armada. Very powerful," Trump told Fox Business Network, adding: "We have submarines. Very powerful. Far more powerful than the aircraft carrier. That I can tell you.”
Referring to North Korean leader Kim Jong Un, Trump said: “He is doing the wrong thing.” Asked if he thought Kim was mentally fit, Trump replied: “I don’t know. I don’t know him.”
North Korea said earlier it was prepared to respond to any US aggression.
"Our revolutionary strong army is keenly watching every move by enemy elements with our nuclear sight focused on the US invasionary bases not only in South Korea and the Pacific operation theater but also in the US mainland," its official Rodong Sinmun newspaper said.
In spite of the military rhetoric, US officials have previously stressed that stronger sanctions are the most likely US course to press North Korea to abandon its nuclear program. At the same time, Washington has said all options - including military ones - are on the table and that a US strike last week against Syria should serve as a warning to Pyongyang.
WATCH: US Navy strike group near Korean Peninsula
The strike group heading toward Korea includes the nuclear-powered flagship aircraft carrier USS Carl Vinson, two destroyers and a cruiser. Such a strike group is generally accompanied by submarines, although the Pentagon does not normally publicize this.
White House spokesman Sean Spicer said Trump had put North Korea "clearly on notice" that he would not tolerate certain actions, but dismissed Pyongyang's nuclear attack threat.
"I think there is no evidence that North Korea has that capability at this time," he said.
"Threatening something that you don't have the capability of isn't really a threat."
North Korea remains technically at war with the United States and its ally South Korea after the 1950-1953 Korean conflict ended in a truce, not a peace treaty. It regularly threatens to destroy both countries.
However, North Korea has conducted five nuclear tests, two of them last year, and is working to develop nuclear-tipped missiles that can reach the United States, presenting Trump with perhaps his most pressing security headache.
WATCH: Tillerson's warning on North Korea
South Korea's acting President Hwang Kyo-ahn warned of "greater provocations" by North Korea, including a possible nuclear test, given a meeting of the country's Supreme People's Assembly and upcoming national anniversaries.
He ordered the military to intensify monitoring and ensure close communication with Washington.
North Korea convened a Supreme People's Assembly session on Tuesday, one of twice-yearly sessions attended by leader Kim Jong Un, and reported a successful national budget execution and personnel appointments, the official KCNA news agency said.
The agency made no mention of North Korea's nuclear weapons program or being under threat from the United States.
Saturday is the 105th anniversary of the birth of Kim Il Sung, North Korea's founding father and grandfather of the current ruler.
A military parade is expected in Pyongyang to mark the day and North Korea often marks important anniversaries with tests of its nuclear or missile capabilities.
Men and women in colorful outfits were singing and dancing on the streets of Pyongyang, illuminated by better lighting than seen in previous years, apparently practicing for the parade.
Trump presses China
Trump said in a Tweet that a trade deal between China and the United States would be "far better for them if they solved the North Korea problem".
"If China decides to help, that would be great," he said. "If not, we will solve the problem without them!"
Trump pressed his Chinese counterpart Xi Jinping to do more on North Korea at a meeting in Florida last week.
China's U.N. ambassador, Liu Jieyi, called the Korean situation "tense" and repeated China's call for a return to dialogue with North Korea.
“We believe that it is highly important to move toward denuclearization, to maintain peace and stability, and it’s time that different sides sit down to talk about achieving these objectives,” he told Reuters.