He reiterated that in the last resolution on North Korea the Security Council stressed its intention to strive for "peaceful and diplomatic solutions" to the situation on the Korean Peninsula and also welcomed the independent efforts of the states to find ways to overcome the crisis.
Tillerson floated the idea of talks without preconditions earlier in the week during a "period of quiet" in North Korean weapons tests.
Tillerson said North Korea must be "held accountable" for illegal detonation of nuclear devices and launching intercontinental ballistic missiles.
Since the spring, the USA has run what it calls a "peaceful pressure campaign" against North Korea.
Tillerson also dismissed concerns that United Nations sanctions are having an impact on North Korea's humanitarian crisis, saying Pyongyang "hypocritically spends billions" on military programs "while its own people suffer great poverty".
"We'd like to have Russia's help - very important", said Trump.
Boris Johnson told fellow Tory MPs that North Korea's rockets would be capable of hitting London by next June.
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. "They alone are responsible for these tensions, they alone must take responsibility for these tensions, and they alone can solve these tensions", he added.
China's deputy United Nations ambassador Wu Haitao said "the core of the nuclear issue is security".
Pyongyang's assertion of ambitions to global supremacy came on Friday after Secretary-General Antonio Guterres warned the Security Council, "The situation on the Korean Peninsula is the most tense and unsafe peace and security issue in the world today". There is "a possibility for negotiation and the option of use of force is unacceptable".
Rising tensions between the United States and North Korea, which conducted its largest nuclear test in September and fired off a powerful ICBM in late November, have raised deep concern worldwide.
The UN's political affairs chief, Jeffrey Feltman, returned at the weekend from talks in Pyongyang - the first visit to the North by a high-ranking UN official since 2011. -China talks and a more comprehensive approach and really an effort to really work on this every day and dare I say maybe fewer press conferences from the secretary of state and more efforts to have serious and prolonged discussions with key allies and partners.
Claiming about its significant backup resources despite the imposed sanctions from China on its military and trading capabilities, Ja said, the UN council is a "tool" of the Americans which is "horrified by the enormous might of our nation".
"A sustained cessation of North Korea's threatening behavior must occur before talks can begin".