President Donald Trump welcomes South Korea's President Moon Jae-In at the White House in Washington, U.S., May 22, 2018.

North and South Korea are discussing a possible non-aggression pledge by the United States to the North and a start of peace treaty talks to address Pyongyang's security concerns before a North Korea-U.S. summit, a senior South Korean official said on Sunday.

South Korean President Moon Jae-in and North Korean leader Kim Jong Un held a surprise second meeting on Saturday after U.S. President Donald Trump called off his talks, set for June 12 in Singapore, before floating a reinstatement of the plan.

"For the success of the North Korea-U.S. summit, we're exploring various ways of clearing North Korea's security concerns at the working level," the senior South Korean presidential official told reporters.

"That includes an end to hostile relations, mutual non-aggression pledge, a launch of peace treaty talks to replace the current armistice," the official said.

The two Koreas are also in talks over a three-way declaration of the end to 1950-53 Korean War but there has not been any agreement yet over a tripartite summit, the official said.