"We will, in the meantime, keep our channels of communication open", Tillerson said.
Last month, North Korea said it had successfully tested a new type of ICBM that could reach all of the U.S. mainland and South Korea and US-based experts said data from the November 29 test appeared to support that. "The pressure campaign must and will continue until denuclearisation is achieved".
The U.N. chief warned that "the risk is being multiplied by misplaced over-confidence, risky narratives and rhetoric, and the lack of communication channels".
The Security Council comprises five permanent members - Britain, China, France, Russia and the United States - and 10 nonpermanent members. As Yun's shuttle diplomacy proceeds, Trump, Tillerson and Defence Secretary James Mattis should coordinate their stances and give the world a clear picture of the direction they want to follow.
Friday's UN Security Council ministerial meeting was convened by Japan, chairing the body this month.
Guterres called on Friday for communication channels between North Korea and South Korea to be restored, as well as between the military in the region, in order to lower the risk of misapprehensions that may lead to a conflict.
Singling out China, Pyongyang's sole ally, and Russia, Tillerson urged Beijing and Moscow to go beyond the current tough United Nations sanctions resolutions and impose unilateral measures. He showed willingness to sit down with the North "with no preconditions".
"There is but one party that has been targeted with punishment and penalties through the most vigorous regime sanction ever enacted, and that is the Kim [Jong Un] regime in North Korea".
He said North Korea won't halt its nuclear and missile programs "while it feels a direct threat to its security" and urged practical steps to de-escalate the situation, including cancellation of planned new U.S.
Trump has also said that "Russia's not helping" and spoke with Russian President Vladimir Putin over the phone Thursday about working together on the situation.
North Koreans "alone are responsible for these tensions, they alone must take responsibility for these tensions, and they alone can solve these tensions", Tillerson said.
China's deputy United Nations ambassador Wu Haitao said "the core of the nuclear issue is security".
He made the comments as he discussed the possibility of the United States pulling its forces from South Korea in a bid to encourage China to overthrown Kim Jong-un.
Some US-based experts believe North Korea could be only two or three tests away from being able to declare the missile operational.
Speaking at a UN Security Council meeting on Friday, the secretary-general noted that he is deeply concerned by "the risk of military confrontation, including as a result of unintended escalation or miscalculation".
"A sustained cessation of North Korea's threatening behavior must occur before talks can begin".