Moon, who had recently returned from his first summit with U.S. President Donald Trump, made the remarks during an informal meeting with former President Barack Obama in Seoul, local news service Money Today reported.
The two men spoke for about 40 minutes. Moon told Obama he and Trump had agreed that sanctions and pressure on North Korea would be accompanied with incentives for dialogue.
North Korea must show cooperative behavior in order for dialogue to take place, Moon said.
Obama told Moon he seeks advice on how to further develop the U.S.-South Korea alliance. He reportedly said the alliance has strong support from the United States and that the "relationship will become stronger."
Yoon Young-chan, a spokesman for the presidential Blue House, said Obama came to Korea in a personal capacity.
Obama also delivered the keynote speech at the eighth annual Asian Leadership Conference in Seoul, the Chosun Ilbo reported.
"The international order depends upon the enforcement of clear rules and norms. So long as North Korea chooses to remain outside of that order, they must face the consequences," Obama said. "We have to insist that the future belongs to those who build, not those who are committed to destroy."
In a separate meeting with Thomas Bach, president of the International Olympic Committee, Moon requested the IOC support North Korea's participation in the Pyeongchang Winter Olympics, Money Today reported.
Bach said he supports Moon's peace policy on the peninsula, according to the report.