On Sunday, Germany, Italy and France urged tougher EU sanctions against North Korea, after Pyongyang reached a new dimension of provocation with its sixth and most powerful nuclear test earlier in the day.
German Chancellor Angela Merkel and French President Emmanuel Macron agreed during a phone call that "North Korea has trampled on international law and that the international community must therefore react with determination against this new escalation."
South Korea says it has seen indications that North Korea is preparing more missile launches, possibly an intercontinental ballistic missile.
It said it was strengthening its controversial US-made Thaad missile defence system after North Korea's test of a nuclear bomb at the weekend.
Seoul has carried out live-fire exercises in response to the test.
The US has warned that any threat to itself or its allies will be met with a "massive military response".
Pyongyang says it tested a hydrogen bomb that can fit on to a long-range missile.
In a statement, British Prime Minister Theresa May said the UN Security Council should urgently look at imposing new sanctions on North Korea and speed up implementation of existing ones.
"This latest action by North Korea is reckless and poses an unacceptable further threat to the international community.
"I discussed the serious and grave threat these dangerous and illegal actions present with Prime Minister Abe in Japan this week and reiterate the call we jointly made for tougher action, including increasing the pace of implementation of existing sanctions and looking urgently in the UN Security Council at new measures."
Further criticism of the regime in North Korea came from European Council President Donald Tusk, who said, "The DPRK's latest nuclear test gravely violates DPRK's obligations under international law, repeated demands from the UN Security Council and the global non-proliferation and disarmament regime. It compels the international community to unite in swift and decisive reaction."
He added, "The DPRK must abandon its nuclear, weapons of mass destruction and ballistic missile programmes in a complete, verifiable and irreversible manner, it must cease all related activities at once."
Tusk went on, "The EU stands ready to sharpen its policy of sanctions and invites North Korea to restart dialogue on its programmes without condition. We call on the UN Security Council to adopt further UN sanctions and show stronger resolve to achieve a peaceful denuclearisation of the Korean peninsula. The stakes are getting too high.
Elsewhere, Nato Secretary General Jens Stoltenberg said he strongly condemned North Korea, saying, "This is yet another flagrant violation of multiple UN Security Council resolutions."
The Norwegian official said, "Nato is concerned by Pyongyang's destabilising pattern of behaviour, which poses a threat to regional and international security. The regime must immediately cease all existing nuclear and ballistic missile activities in a complete, verifiable, and irreversible manner, and re-engage in dialogue with the international community."
Stoltenberg said he had urged North Korea to "respect its international obligations, and to abandon all threats and actions which contribute to tension and insecurity."
In the past two months North Korea has conducted intercontinental ballistic missile tests, sending one over mainland Japan into the Pacific Ocean. It has also threatened to fire missiles towards the US Pacific territory of Guam.
The United Nations Security Council is to hold an emergency meeting later on Monday to discuss its response.