South Korean President Moon Jae-in said it is only a matter of time before the US and North Korea declare an end to their state of war on the Korean peninsula.
The war ended in 1953 with an armistice but a peace treaty was never signed.
Mr Moon, the son of North Korean refugees, also said there could be more diplomatic "bumps and bruises" as he tries to persuade Kim Jong-un to give up his nuclear weapons.
But he, in an interview with BBC, described Mr Kim as "candid".
He also said he hoped European leaders would help him to mediate between Mr Kim and US President Donald Trump if negotiations stalled.
Mr Moon has met Mr Kim three times this year and has acted as a mediator between him and Mr Trump.
Mr Moon said that he had had "ample discussions" with President Trump and other US officials about an end-of-war declaration.
"If North Korea takes certain measures, the end-of-war declaration would be a political statement that would announce that the longstanding hostile relations between Pyongyang and Washington had ended," he said.
Mr Moon said he wanted this to happen at the "earliest possible date" and he believed there was a "shared understanding between Washington and Seoul regarding this view".
Last month, he became the first South Korean leader to give a speech to the North Korean public during a trip to Pyongyang. He spoke at the Arirang Games - a major propaganda event - and received a standing ovation from 150,000 people.
"I was actually quite nervous to give the speech," he said. "I had to address the problem of denuclearisation and had to receive positive vibes from the North Korean people. And also I had to satisfy the Korean public as well as the global public. So it was not an easy task for me."
© 2017 - All Rights with The Financial Express