SEOUL—North Korea is open to allowing outside inspections of a nuclear-weapons testing site it closed in May, South Korea’s leader said Thursday, just a day after the North agreed to open a missile site to inspectors.

Back from a three-day summit with North Korean leader Kim Jong Un, President Moon Jae-in said Mr. Kim is willing to allow inspection of the site, Punggye-ri, which wasn’t mentioned in this week’s joint agreement signed by the two countries.

“The...