As if to underline Seoul's willingness to be tough, the South Korean military conducted bombing drills at dawn Monday, practicing ballistic missile strikes on the North Korean nuclear test site at Punggye-ri.
Treasury Secretary Steven Mnuchin said that "no decisions" had been made but that trade deals must be in the United States' economic interest.
Reports noted that South Korea had previously asked the U.S.to review the missile guidelines in a bid to double its firepower following rising threats from the North.
Tuesday: The U.S. tests missile defense in HawaiiThis US Navy handout photo released on August 30, 2017 shows a medium-range ballistic missile target as it is launched from the Pacific Missile Range Facility in Kauai, Hawaii, during Flight Test Standard Missile-27, Event 2 on August 29, 2017.
It came as USA and South Korean forces were nearing the end of the 10-day annual Ulchi Freedom Guardian joint exercises, which the North regards as a rehearsal for invasion.
The latest step comes after a roller-coaster month of escalating tensions over North Korea.
May's first trip to Japan as prime minister comes at a hard moment, with Japanese attention focused on North Korea's Tuesday missile test and Britain's Brexit negotiators recently on the receiving end of sharp criticism from European Union officials.
At a press conference alongside Japanese counterpart Shinzo Abe, the Prime Minister said the United Kingdom condemned the latest missile launch in the "strongest terms possible".
Trump also gave "conceptual" approval to the purchase by the South of billions of dollars of USA military hardware, the White House said.
But a possible rotational deployment reflects growing concerns about a fast advance in North Korea's nuclear and missile programs. Washington, in turn, offers a "nuclear umbrella" against potential attacks. And repeated rounds of United Nations resolutions and sanctions against North Korea appear to have had little or no impact.
The drills were a clear warning after North Korea launched a midrange ballistic missile created to carry nuclear bombs over Japan earlier this week, South Korea's military said.
Sources confirmed Song told McMaster there are calls in South Korea for the deployment of nuclear weapons.
While no details have yet emerged regarding what the new resolution would contain, South Korea has also expressed its support for the passage of new measures - as with May - in conjunction with the government of Japan.
Abe called the launching of a North Korean missile over Japan earlier this week an unprecedented and outrageous act.
In his tweet, Trump declared: "The US has been talking to North Korea, and paying them extortion money, for 25 years".
It also appeared to be the North's longest-ever missile test, but South Korean officials couldn't immediately confirm. A Standard Missile-3 Block IIA jointly developed by the USA and Japan failed to intercept its target. Some experts believe the next North Korean launches will be bolder unless Washington makes serious concessions.
Tudor has not been to North Korea since publication and is understood not to have plans to go again.