Washington has long insisted the North must show a willingness to denuclearise before any negotiations - which Pyongyang says it will never do.
The North Korean taekwondo athletes arrived South this week and performed at the Pyeongchang Olympic Games opening ceremony on Friday as part of four performances during their nine-day trip.
Moon also encouraged the North Koreans to be more active in pursuing talks with the United States, saying that "early resumption of dialogue (between the two) is absolutely necessary for developments in the inter-Korean relations as well".
The recent detente, anchored by South Korea's hosting of the Winter Olympic Games that began on Friday, came despite an acceleration in the North's weapons programmes a year ago and pressure from Seoul's allies in Washington.
The team took to the ice for the first time on Saturday, losing to Switzerland but winning over the crowd as North Korean cheerleaders and South Korean fans roared the players on.
The officials said it was fair to say both Pence and the North Koreans rebuffed South Korean President Moon Jae-in's efforts to facilitate a meeting between North Korea and the US.
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.
Facebook 'losing youth at fast pace', US study shows
In detail, the eMarketer analysis shows Facebook will lose around 2 million users aged 24 and under, during 2018. Meanwhile, more over-55s are signing up, with half a million new users expected to join by the end of the year.
Moon and Kim Yo Jong had their first face-to-face encounter at the opening ceremony.
Kim Yo Jong cuts a contrasting figure to her brother in North Korea's formal and tightly controlled state propaganda.
"We're going to continue to put all the pressure to bear economically and diplomatically, while preserving all of our military options to see that that happens", Pence told NBC.
North Korean leader Kim Jong Un presided over an extravagant military parade in Pyongyang's Kim Il Sung Square on Thursday, grabbing the spotlight on the eve of the Pyeongchang Winter Olympics in South Korea to thumb his nose at Washington while making a point of showing off his new-found restraint toward Seoul.
The sight of North and South Koreans marching together for the first time in a decade was the culmination of months of work by Seoul.
The north's state-run agency said the talks were "warm" but added they were "frank and candid". She also asked him to play a "leading role" in reuniting the two Koreas after almost seven decades.
Moon's desire to engage North Korea was in contrast to his United States ally.