Under the leadership of Kim, North Korea has been speeding up its pursuit of a decades-long goal of developing nuclear-armed missiles capable of reaching the US mainland.
He also said the North's nuclear and missile capabilities seem to have advanced rapidly recently but that the South was ready and capable of striking back should the North attack.
President Donald Trump is looking to both China and Russian Federation, the two permanent members on the Security Council that have historically been most sympathetic to North Korea, to join the US -backed campaign of diplomatic and economic pressure on the North to get it to denuclearize.
The two countries also confirmed that Moon and Trump shared four broad principles in dealing with the North Korean nuclear crisis, Yoon said.
South Korean President Moon Jae-in, who took office last week, has campaigned on a more moderate approach towards the North but he has said it must change its attitude of insisting on arms development before dialogue can be possible.
This is the first time a US president received a South Korean special envoy in the Oval Office, and the unexpected attendance of Trump's son-in-law Kushner in the meeting was also seen as a sign of respect for the Seoul delegation.
Hong, chairman of the Korean Peninsula Forum and a former USA ambassador, led the Korean delegation to Washington Wednesday, the first meeting of an envoy of the new Moon administration with Trump. Hong said South Korea had not yet received official word from the United States on whether Seoul should pay for an anti-missile US radar system that has been deployed outside Seoul.
Seoul has complained that some South Korean companies doing business in China have faced discrimination in retaliation for the system's deployment.
Moon has said that he would meet with the North Korean leader, Kim Jong Un, if the circumstances were right.
The United Nations Security Council first imposed sanctions on North Korea in 2006 and has strengthened the measures in response to its five nuclear tests and two long-range rocket launches.
Before an emergency meeting of the council Tuesday, U.S. Ambassador Nikki Haley declared: "You either support North Korea or you don't, but you have to choose".
While Moon may try to soothe Chinese anger over the THAAD deployment, many observers say it appears too late to call for its withdrawal, which would greatly undermine South Korea's relationship with Washington, its most important ally.
South Korea's new president launched worldwide efforts to defuse tension over North Korea's weapons development on Thursday, urging both dialogue and sanctions while also aiming to ease Chinese anger about a United States anti-missile system.