"The council must be prepared to impose additional and stronger sanctions on North Korean nuclear and ballistic missile programmes," he said.
US officials have told Reuters tougher sanctions could include an oil embargo, banning North Korea's airline, intercepting cargo ships and punishing Chinese banks and other foreign doing business with Pyongyang.
Trump said North Korea "is a real threat to the world".
"North Korea is a big world problem and it's a problem that we have to finally solve," he said. “People put blindfolds on for decades and now it’s time to solve the problem."
The comments came as Japan conducted exercises with a US aircraft carrier strike group headed for Korean waters.
Angered by the approach of the USS Carl Vinson carrier group, a defiant North Korea, which has carried out nuclear and missile tests in defiance of successive rounds of UN sanctions, said the deployment was "an extremely dangerous act by those who plan a nuclear war to invade".
"The United States should not run amok and should consider carefully any catastrophic consequence from its foolish military provocative act," Rodong Sinmun, the official newspaper of North Korea's ruling Workers' Party, said in a commentary.
South Korean and US officials have feared for some time that a sixth North Korean nuclear test could be imminent and speculation has grown that this, or another missile test, could coincide with the 85th anniversary of the foundation of North Korea's army on Tuesday.
But South Korean defence officials said no suspicious moves by North Korea were detected on Tuesday morning.
Trump has vowed to prevent North Korea from being able to hit the United States with a nuclear missile and has said all options are on the table, including a military strike, although officials say tougher sanctions are the preferred route.
Envoys on the North Korean nuclear issue from the United States, South Korea and Japan are meeting in the Japanese capital, Tokyo, on Tuesday.
Al Jazeera's Scott Heidler, reporting from Tokyo, said no policy change announcements are expected from the meeting, which is an opportunity for the three countries to plan and coordinate their response to any potential action by North Korea.
"It's not unprecedented that they sit down like this, but obviously with this current situation, it's much more important," he said.
Source: Al Jazeera and news agencies