SEOUL—The U.S. and North Korea have been engaged in informal talks about a third nuclear summit between the two countries’ leaders, South Korean President Moon Jae-in said, signaling an interest by both sides in breaking a monthslong lull in negotiations.

Mr. Moon didn’t provide a timeline for the potential meeting between President Trump and North Korean leader Kim Jong Un. He said the two sides had held “behind the scenes” talks, not official dialogue, after a February summit in Hanoi ended abruptly without a deal.

The...