A White House spokesman added that Mr Moon had told Mr Pence no one would take pressure off North Korea until it takes steps to denuclearisation.
Earlier, it has suggested that talks could start between the USA and North Korea once South Korea suspends its joint military exercise with South Korea.
"We surely have to win competitions but I mean attending the Olympics has a significant meaning", the North's alpine coach, Pyon Yong Do, said after a meeting of coaches on Sunday.
Last year, North Korea conducted dozens of missile launches and its sixth and largest nuclear test in defiance of U.N. Security Council resolutions as it pursues its goal of developing a nuclear-armed missile capable of reaching the United States.
Pence endorsed a change of tactics after Moon assured him the North Koreans wouldn't get economic or diplomatic benefits just for talking-they must take concrete steps toward denuclearization.
The U.S., on the other hand, led by Trump and Pence, isn't making much of an effort to embrace all the recent attempts at diplomacy with North Korea.
North Korea's delegation to the games will conclude its visit later on Sunday, having appeared to thaw the rift with the South, but there is scepticism about how honest Pyongyang is about improving relations.
'Too soon' to tell if Korean Olympic thaw will last: Mattis
Her job is to pump out propaganda both venerating her brother and the North Korean state and discrediting enemies. Afterwards, I made three more attempts, but every time I was forced to be sent back to North Korea.
Kim Jong-un is merely trying to use the South Korean government as a shield to protect him against intensifying worldwide pressure, and he thinks the most effective way to do that would be to hold an inter-Korean summit.
Moon is expected to study the North's call for a bilateral summit, while monitoring whether Pyongyang will move to have dialogue with the United States as requested by his government.
The protesters set the flag alight along with a portrait of Kim Jong Un, whose sister was due to watch a Northern musical display with the South's President at a nearby theatre in Seoul.
Vice president Mike Pence said in a newspaper interview the US and South Korea have agreed on terms for further diplomatic engagement with North Korea, first with Seoul and then possibly leading to direct talks with Washington without pre-conditions.
As Moon himself has pointed out, the inter-Korean dialogue achieved this time was like a miracle.
South Korea yesterday investigated a mysterious internet shutdown during the Winter Olympics opening ceremony, which follows warnings of possible cyber attacks during the Pyeongchang Games.
Speaking about the possibility of meeting with North Korean officials, U.S. Vice President Mike Pence said that he would "wait and see" but use any opportunity to pressure Pyongyang to abandon its nuclear ambitions.
South Korea said it will seek ways to continue engaging North Korea including trying to arrange more reunions for families divided by the Korean War and aim to lower military tensions.
The younger Kim's trip to the South marks the first time that a member of the North's ruling dynasty has visited since the Korean War, which ended in an armistice in 1953. He also suggested that North Korea should engage in talks with the United States.