The contacts are occurring regularly between Joseph Yun, the USA envoy for North Korea policy, and Pak Song Il, a senior North Korean diplomat at the country's United Nations mission, according to U.S. officials and others briefed on the process.
The President said America would "of course" always consider negotiations with North Korea, but added that talks had failed for the last 25 years.
Asked whether Mr Trump was wise to threaten to unleash "fire and fury" in retaliation, Mr Green said: "I think the sensible way for people to proceed is to work through the United Nations process, that's what the British Government has been supporting and will continue to support".
Trump added that North Korean leader Kim Jong-un has "disrespected our country greatly", adding, "with me, he's not getting away with it".
The exercise began hours after Japan's defense minister said the Japanese military could shoot down North Korean intercontinental ballistic missiles before they reach the US territory of Guam. He also spoke of pursuing more sanctions on North Korea.
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"Trump's bombastic rhetoric is not appropriate when we are dealing with the possibility of a nuclear war that could kill millions of people", the senator from Vermont tweeted.
The government has chose to deploy Patriot interceptor missiles in parts of western Japan that North Korea's missiles are expected to fly over if Pyongyang actually goes through with its plan to fire them at Guam, Defense Ministry officials said Friday.
"Ever since I served [in the] military ten years ago, (the North Korean threat) has been continuing, and South Korean men who served in the military have developed an understanding about North Korean provocations, so we get used to it", said Seoul resident Jang Min-seok.
On Thursday Mr Trump suggested that his own statements on North Korea had not been tough enough, warning the regime to be "very, very nervous".
It said the President's warning was a "load of nonsense" and threatened a fully developed plan for striking Guam with missiles by mid-August.
Tensions have escalated since North Korea tested two intercontinental ballistic missiles in July. He also retweeted a US Pacific Command message that said US planes stand ready to "fulfill a #FightTonight mission".
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The annual joint U.S. "If there is an attack on Australia or the United States, then each of us will come to the other's aid".
Japan and South Korea vowed a strong reaction if the North were to go through with the plan. You are talking South Korea, the 11th largest economy in the world, Japan, the third largest economy, and you are talking about ground troops on the Korean peninsula.
By North Korea's boastful reckoning, its Hwasong-12 missiles will take just 1065 seconds to complete their 3356km flight over Japan before splashing down in waters off the coast of Guam.
First Secretary of State Damian Green said it is "obviously" in Britain's interests that the stand-off between Washington and Pyongyang does not lead to conflict. While critics express concern about the increased pace of North Korean missile tests, NBC's sources said the launches have helped them refine their understanding of the communist state's missile facilities.
The United States and South Korea remain technically at war with North Korea after the 1950-53 Korean conflict ended with a truce, not a peace treaty.
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