"In the midst of all of this, he (North Korean leader Kim Jong Un) ran a military parade that highlighted his ballistic missiles", Mattis told reporters on his plane en route to Rome on the first leg of a European tour.
Any meeting would represent a diplomatic coup for Moon, who swept to power past year on a policy of engaging more with the reclusive North.
U.S. Defense Secretary Jim Mattis said on Sunday it was too early to tell whether a diplomatic detente between the two Koreas at the Winter Olympics would lead to results, but the move had not driven a wedge between Washington and Seoul.
"Special envoy Kim Yo Jong delivered a personal letter" from her brother stating his "wish to improve inter-Korean relations", said Moon's spokesman Kim Eui-kyeom.
The Moon government has been engaging in a delicate balancing act between Pyongyang and its longtime ally Washington after Kim Jong Un offered to send a delegation to the Olympics in the South. Moon has accommodated the North's demands on Olympics participation in hopes of persuading Pyongyang to return to the negotiating table to discuss denuclearization, while trying to allay concern in Washington that the North was using a "charm offensive" to simply ease sanctions and earn time to complete its nuclear weapons program.
Some North Korean experts believe tough United Nations sanctions that are cutting off most of the isolated North's sources of revenue have added pressure on Pyongyang to engage further with Seoul.
The US is ready to engage in direct talks with North Korea even as it maintains its "maximum pressure campaign" on the reclusive state, US Vice President Mike Pence said, hinting at a shift in the US' North Korea policy.
Moon asked the North Korean delegation during Saturday's meeting to engage in dialogue with Washington "at an early date", the Blue House said.
Moon's desire to engage North Korea was in contrast to his US ally.
Pence also attended the Games opening ceremony but had no contact with the North Korean delegation.
North and South Korea are technically still at war after their 1950-53 conflict ended in a truce rather than a peace treaty.
When he insisted to Mr Moon that South Korea and the United States should resume their joint Foal Eagle military exercise after the Olympics, he was told not to meddle in the South's "sovereignty and internal affairs".
North Korea has spent years developing its military, saying it needs to protect itself from United States aggression.
But analysts say it may be more hard for the South to arrange a summit with the North coming off a year in which Pyongyang test-fired dozens of missiles, including three ICBMs, and conducted its most powerful nuclear test to date. The U.S. -led worldwide community has been tightening the screws on North Korea with sanctions created to punish its economy and rein in its efforts to expand its nuclear weapons and missile program, which now includes developmental long-range missiles targeting the U.S. mainland.
It's by far the most sustained public presence for Kim, who had been something of a mystery to many South Koreans, despite her known role in North Korea's politburo.
"Under a strong position for denuclearisation of the Korean peninsula, Korea will faithfully implement the worldwide sanctions on North Korea, while also adhering to the principle of resolution through peaceful means", the statement said.
Are these Winter Games having the effect South Korea had hoped, in thawing relations with the north?
The two Koreas have a rocky and sometimes violent history at the Blue House.
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