He adds that at some point he'd like to invite Kim to the United States. He said, "We are getting along very well".
Moon's spokesperson Kim Eui-Kyeom appreciated Trump's comment on second meet with Kim and said, "We hope a concrete agenda and logistics will be determined soon".
In that historic summit, Trump and Kim opened up dialogue on the denuclearisation of the Korean peninsula after trading military threats for months.
On Friday, a South Korean train left for North Korea, launching an 18-day inspection of railways there as part of efforts to eventually connect rail lines over the border, a project their leaders agreed on earlier.
When asked about a future meeting, Mr. Trump said: "I think we're going to do one fairly [soon] - into January, February, I think".
Pence has also said it was essential that worldwide sanctions pressure be maintained on North Korea until its complete denuclearisation was achieved.
North Korea has sought security guarantees from the USA and relief from worldwide sanctions.
U.S. President Donald Trump and North Korea's leader Kim Jong Un walk together before their working lunch during their summit at the Capella Hotel on the resort island of Sentosa, Singapore June 12, 2018.
The last defection involving North Koreans occurred in May when two civilians aboard a small boat fled to the South across the Yellow Sea.
President Moon Jae-in's office quoted Moon as saying that Trump asked him to convey those messages to the North Korean leader if he visits Seoul this year as he promised. After their third meeting in Pyongyang, North Korea's capital, in September, Moon said Kim agreed to make a reciprocal visit to Seoul this year.
Trump "hopes for Chairman Kim Jong Un to fulfill the rest of his agreements, and said he will grant what Chairman Kim Jong Un wants", Moon told reporters.
Former UN Secretary-General Ban Ki-moon has urged North Korean leader Kim Jong Un to take concrete steps toward complete denuclearization and gain the trust of the worldwide community if he wants sanctions lifted.