In a phone call Friday, Mr. Trump praised Chinese President Xi Jinping for supporting the United Nations vote, and the leaders reaffirmed their commitment to a nuclear-free Korean peninsula.
Dunford was on the first stop of a trip that will also take him to Beijing Tuesday and then on to Tokyo, three capitals that do not want war to break out on their doorsteps.
North Korea has invited a limited attack on it's testing facilities by public announcement of intent to do testing which in and of itself will threaten possible harm to the citizens of other countries.
Mogherini called on North Korea "to refrain from any further provocative action that can only increase regional and global tensions".
The ban will take effect on Tuesday, August 15, excluding products that have been shipped to ports and are awaiting domestic transportation.
South Korean President Moon Jae In called Monday for calm in the standoff with North Korea, saying there should never be another war on the peninsula. Kim Il-Sung, leader of North Korea, was a former guerrilla who fought under Chinese and Russian command.
White House officials have insisted the investigation was not created to apply additional pressure on China as it relates to North Korea despite the president's previous remarks that he would be more amenable on trade if China stopped Pyongyang's nuclear program. We must act now to prevent the prospect of a nuclear war.
"There's nothing imminent today", he said on "Fox News Sunday".
The Defence Secretary Mr Mattis and Secretary of State Rex Tillerson have also penned an opinion piece in the Wall Street Journal.
Trump's posture suggested he was trying to dissuade Kim from further provocations rather than setting the stage for a U.S. military strike, according to Terence Roehrig, a national security affairs professor at the U.S. Naval War College in Newport.
"North Korea best not make any more threats to the United States. Whatever ups and downs we face, the North Korean nuclear situation must be resolved peacefully", Mr Moon told a meeting of senior aides and advisers.
Dunford "stressed that North Korea's ballistic missile and nuclear weapons programs threaten the entire global community", according to United States military spokesman Capt. Darryn James.
About one-third of the USA territory's 160,000 people identify as Chamorro, the indigenous group that is believed to have migrated to Guam from Indonesia and the Philippines an estimated 3,500 to 4,000 years ago.
"I call upon the North Korean government: without worldwide cooperation and co-existence economic development is impossible", said Moon.
The strong USA military presence on Guam, located 2,000 miles (3,200 kilometers) southeast of Seoul, has left it the target of North Korea's threats for years.
Pyongyang has warned that even "a small accidental event" during the drills could make a nuclear war unstoppable. "We are closely monitoring every move by the United States".
North Korean leader Kim Jong Un reviewed his military's plans to rain "an enveloping fire" around the US territory of Guam - but opted not to fire missiles at this time, according to state media.
The Catholic faithful in Guam led prayers for peace Sunday in the shadow of North Korean missile threat, with the western Pacific island's archbishop appealing for "prudence" amid an escalating war of words between the USA and Pyongyang.
The Korean Central News Agency kept up its tough talk Monday.