In a telephone call on Sunday, Trump's national security adviser, H.R. McMaster, reassured his South Korean counterpart, Kim Kwan-jin, that the US alliance with South Korea was its top priority in the Asia-Pacific region, the South's presidential office said.
A ballistic missile launched early Saturday by North Korea in defiance of global pressure and at a time of heightened regional tensions appears to have failed.
Chief Cabinet Secretary Yoshihide Suga said the ballistic missile was sacked to the northeast around 5.30am from an area in the vicinity of Pukchang, north of Pyongyang, the capital.
President Donald Trump would not rule out the use of military force against North Korea in an interview on CBS's "Face the Nation" on Saturday, just hours after Pyongyang launched a missile test in defiance of worldwide pressure.
US President Donald Trump did not remain silent after the test launch, tweeting: "North Korea disrespected the wishes of China & its highly respected President when it launched, though unsuccessfully, a missile today".
South Korea said the United States had reaffirmed it would shoulder the cost of deploying the THAAD anti-missile system, days after President Donald Trump said Seoul should pay for the $1-billion battery created to defend against North Korea.
US and South Korean officials said the launch apparently failed.
The launch was swiftly condemned by South Korean and Japanese leaders.
In a speech to the UN Security Council on April 28, Tillerson called on UN members to strengthen financial sanctions against North Korea or downgrade their diplomatic relations with Pyongyang in response to the regime's nuclear and ballistic-weapons programs.
"The U.S. National Security Adviser H.R. McMaster explained that the recent statements by President Trump were made within the context in line with the general U.S. public expectations on burden-sharing with allies", a South Korean presidential office spokesperson told NBC News.
He said diplomatic efforts are preferred, but he also expressed the willingness of the United States to take military action, if necessary, to counter North Korean aggression.
Trump, in his interview with Reuters, said he had praised Chinese leader Xi Jinping for "trying very hard" on North Korea but warned a "major, major conflict" between the United States and North Korea was possible.
"We will not negotiate our way back to the negotiating table", he said.
In a further show of force, the USS Carl Vinson aircraft carrier group arrived in waters near the Korean peninsula and began exercises with the South Korean navy late on Saturday. The missile launch comes amid high tensions in the Korean peninsula.
It is not clear what has caused the series of failed missile tests since then.
Moon has advocated a more moderate policy on the North and been critical of the deployment of an advanced US missile defense system in the South meant to counter North Korea's missile threat, which China also strongly objects to.
North Korea recently conducted its biggest-ever firing drill and has threatened to "bury at sea" the USA aircraft carrier, and there are signs it could be preparing for a sixth nuclear test. Some of those missiles reached the Sea of Japan, also known as the East Sea.
But UN sanctions experts have repeatedly told the council the measures have had little impact because they have been poorly implemented.
With less than two weeks until South Korea votes for a new president, the spokesman for the frontrunner, Moon Jae-in, called on North Korea to stop its military tests.
He said China was "in communication with the regime in Pyongyang" and had "confirmed to us that they had requested the regime conduct no further nuclear tests".