President Moon Jae-in has previously floated the idea of a summit with Kim under conditions. But his latest comments came a day after the North agreed to send a delegation to next month's Winter Olympics in South Korea and hold military talks on reducing animosities along their tense border, the measures that Moon's government has been demanding.
The accords, reached at the rivals' first meeting in about two years, were widely viewed as a positive step following a year of escalating tension over Kim's advancing nuclear and missile programs.
They also say Kim may push for better ties with Seoul as North Korea feels the pain of US-led international sanctions.
During a televised news conference in Seoul, Moon described the North's Olympic participation as "very desirable," but said that inter-Korean relations cannot be improved without progress in the nuclear standoff and vice versa.
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Moon said North Korea will face harsher international sanctions and pressure if it resorts to fresh provocations, adding that "denuclearization of the Korean Peninsula is path to peace and our goal."
"To have the summit, some conditions must be established. I think a certain level of success must be guaranteed," Moon said.
US State Department spokeswoman Heather Nauert welcomed yesterday's Korean talks, which she said were "aimed at ensuring a safe, secure and successful" Olympics. White House spokeswoman Sarah Huckabee Sanders also said the North's decision to attend the Games presents it with an opportunity to see the value of ending its isolation from the rest of the world.
Trump recently also contended his tough words helped persuade the North to sit down with the South. During Wednesday's conference, Moon said he feels thankful to Trump for making the talks happen.
Under Thursday's deal struck at the Koreas' border village, North Korea will send a delegation of officials, athletes, cheerleaders, journalists and others to Pyeongchang. The accord stipulates the two Koreas would "actively cooperate" in the Olympics to "enhance the prestige of the Korean people."
During the talks, North Korea told the South Korean delegation that they recently restored a military hotline with the South in the second such reopening of an inter-Korean network, according to South Korean officials.
All major communication had been shut down over the North's nuclear program in recent years. But North Korea reopened one channel last week as signs emerged of improving ties.
But animosities flared again several months later after the North's fourth nuclear test.
Last year, North Korea conducted its sixth and most powerful nuclear test and three tests of intercontinental ballistic missiles, and Trump and Kim exchanged bellicose rhetoric and crude insults against each other.