South Korean President Moon Jae-in and US President Donald Trump on Monday discussed possible ways to reward North Korea for its denuclearisation measures that will apparently include a second US-North Korea summit.

"The leaders agreed to continue communicating closely about corresponding measures," said Kim Eui-kyeom, a spokesman for South Korea's presidential office Cheong Wa Dae, S. Korean News Agency (Yonhap) reported.

In their bilateral summit held in New York, the leaders of South Korea and the United States also agreed on the need to show North Korea a bright future following its denuclearisation.

"The leaders agreed to seek ways to encourage North Korea's denuclearisation process by showing a bright future it may enjoy once it reaches complete denuclearisation," the Cheong Wa Dae spokesman said.

The leaders noted the need to maintain international sanctions against the North for the time being, he added.

Moon arrived Sunday for a four-day trip that follows his three-day visit to North Korea last week for his third bilateral summit with North Korean leader Kim Jong-un that was largely aimed at breaking a deadlock in denuclearisation talks between the US and North Korea.

Trump and Kim held the first-ever US-North Korea summit in Singapore in June. The US president said the date for his second meeting with Kim will be announced soon, adding the meeting will likely be held in a place other than Singapore.

After the summit, the spokesman said Moon and Trump had in-depth discussions on the date and location of Trump's second meeting with Kim, possibly suggesting the leaders may be considering a three-way summit involving Moon.