The conversation followed another North Korean missile test-launch on Saturday, which Washington and Seoul said was unsuccessful.
Seoul and the United States say the sole objective of THAAD is to guard against North Korean missiles. Dave Benham, a spokesman for the US Pacific Command.
So with military action and diplomatic strategies both at work here, what are the circumstances under which Trump would actually be willing to sit down with Jong Un?
The launch was swiftly condemned by South Korean and Japanese leaders.
China is "putting pressure" on its ally North Korea to curb its weapons programmes, Trump told the CBS television network's "Face the Nation" programme.
The exercise, called Foal Eagle, was repeatedly denounced by North Korea, which saw it as a rehearsal for war.
The THAAD system in South Korea has reached an initial operating capability to defend against North Korean missiles, US officials said on Monday. Fight with North Korea? Unlike the United States, Beijing has pushed for talks first and action later on North Korea. Say things to provoke and then find some way to justify them.
But White House spokesman Sean Spicer says he doesn't see that happening anytime soon. We're probably not safe over here. The KN-17 is a land-based solid-fuel missile fired from a mobile launcher. "We need cooperation at some level with as many partners in the area as we can get to make sure that we have our ducks in a row", Priebus told ABC's "This Week".
There has been no announcement on North Korean state television, CNN's Will Ripley in Pyongyang reported.
The country's nuclear program, along with its numerous ballistic missile tests, have made containing North Korea the top foreign priority of Mr. Trump's administration.
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They had little interest in the in-fighting between his senior aides in the White House. "I like to drive", he said. Others lament he is not pulling America from global conflicts as he vowed - or say he should "get off of Twitter".
President Donald Trump told Bloomberg Monday that he would be "honored" to meet with North Korean dictator Kim Jong Un under the right circumstances, the latest in a series of odd moves that suggest Trump may still not grasp the power and signal-sending of such a gathering.
He has so far remained tight-lipped about his strategy with dealing with North Korea's ballistic missile testing.
But "there is reason to be concerned" about North Korea's missile tests, she added.
Trump said in an interview with Bloomberg News.
Washington is hopeful the Chinese can help. But in an interview Sunday with Fox News, McMaster denied that suggestion.
And at such a unsafe time, Cirincione is trying to ensure that it won't be Trump alone who will have the right to trigger an American nuclear response against North Korea.
China remains one of North Korea's only allies and is responsible for much of the heavily-sanctioned nation's economy.
"The THAAD interceptor is very easily defeated by either causing a missile to tumble end over end or by intentionally fragmenting a rocket into pieces" a symbolic remark by Theodore Postol, a professor at the Massachusetts Institute of Technology, to the heavy critique that was thrown at the project of the pentagon.
Yet on other occasions, Trump's administration has dangled carrots. Some of those missiles reached the Sea of Japan, also known as the East Sea. The vote also has become a referendum on USA relations - about how close South Koreans want to be with the United States.