Last week the North said it may cancel the summit if the US insists on it giving up nuclear weapons unilaterally.
Mr Moon will try to calm US nerves to get talks back on track, observers say.
One of his officials said he would “likely tell President Trump what to expect and what not to expect from Kim”, Yonhap news agency reported.
The historic 12 June summit is due to take place in Singapore.
What has North Korea said?
The summit was first thrown into doubt when North Korea cancelled high-level talks with South Korea last Wednesday, saying it was angered by Seoul’s joint-military exercises with the US.
The North called the exercises a “provocation” and a rehearsal for an invasion.
Pyongyang was then highly critical of US National Security Adviser John Bolton, whom it accused of making “reckless statements” after he suggested the North could follow a “Libya model” of denuclearisation.
That was a reference to Libya’s former leader Muammar Gaddafi who agreed to give up nuclear weapons in 2003. He was later killed by Western-backed rebels.
What has been the US response?
Mr Trump later clarified that the US was not pursuing the “Libya model” for North Korea.