The talks in the village of Panmunjom, in the demilitarised zone between the two Koreas, are the latest sign that the summit could take place after all.
On Thursday Mr Trump called off the meeting - due in Singapore next month - citing the North's "hostility".
But both sides have since been working to get it back on track.
On Saturday Mr Kim and South Korean President Moon Jae-in held an unannounced meeting. Mr Moon said the North's leader had "again made clear his commitment to a complete denuclearisation of the Korean peninsula".
Mr Kim later spoke of his "fixed will" that the summit should go ahead. On Sunday Mr Trump tweeted that it would help the North achieve its "brilliant potential".