The reports came after Trump said that the North Korean nuclear issue would be "taken care of", while at the same time calling on China to pressure its neighbor into giving up its nuclear program.
The United States will enhance its military presence in the Asia Pacific region amid growing threat emanating from North Korea, US Vice President Mike Pence said Wednesday.
Experts point out that China also wants to prevent North Korea from becoming a full-fledged nuclear power - and certainly wants to prevent a war on its southern border that could send millions of refugees flooding into China and potentially risk bringing a United States military presence to China's borders.
He ensured military personnel that under Donald Trump "the shield stands guard and the sword stands ready".
US Secretary of State Rex Tillerson said the United States was looking at ways to pressure North Korea over its nuclear programme as North Korean state media warned the Americans of a "super-mighty preemptive strike", saying don't "mess with us".
Graham said he recommended Trump tell China that if Beijing doesn't stop Pyongyang from building a missile capable of striking America, the United States "will do everything, including diplomatic sanctions and military strike to stop the missile program".
Mr Pence made the statement after talks in Tokyo with the Japanese Prime Minister Shinzo Abe.
The term "reckless" is one the North Koreans have used to describe continuing large-scale US and South Korean military exercises, which the North calls a dress rehearsal for an invasion.
"I think it's important that North Korea knows we're not trying to pick a fight, so don't try and give us one", she said.
Another official, speaking on condition of anonymity to discuss an intelligence matter, said the missile was a Scud variant that the USA calls a KN-17.
Abe said that it was a "matter of paramount importance for us to seek diplomatic efforts as well peaceable settlements of the issue".
The North has said such launches are an act of deterrence against a potential invasion by the US or South Korea.
"China expresses serious concern with recent trends about North Korea's nuclear and missile development".
Foreign ministry spokesman Lu Kang said China opposed words or actions that could further raise tension.
"China is unswerving in its commitment to realising the denuclearisation of the Korean peninsula, maintaining the peace and stability of the Korean peninsula, and continue to solve matters through dialogue and negotiations".
Roughly 1,000 American service members and about 500 South Korean troops will carry out Operation Max Thunder, a training exercise involving aircraft carriers, fighter jets and military drills.