This week of reunions was one of the main agreements made by South Korean President Moon Jae-in and North Korean leader Kim Jong Un at their April summit.
Ahmad Khatami also told worshippers attending Eid prayers in Tehran that President Donald Trump's offer of talks with Iranian leaders was unacceptable as the usa leader wanted Tehran to concede on its missile programme and regional influence.
In the interview, Trump said he had developed "very good personal relations" with Kim at their June summit in Singapore.
Trump said he believes the North Koreans have taken concrete steps to denuclearize beyond demolishing the nuclear test site at Punggye-ri, a measure critics have derided as mere "grandstanding".
White House National Security Adviser John Bolton met Israeli Prime Minister Benjamin Netanyahu in Jerusalem on Sunday evening, telling the premier that Iran's nuclear and ballistic missile programs were "right at the top of the list" of challenges facing the United States and the world, The Times of Israel reported.
In the interview, Trump pointed to North Korea's suspension of nuclear and ballistic missile testing and took credit for making it happen.
The emotional reunion came after dozens of elderly South Koreans crossed the heavily fortified border into North Korea to meet temporarily with their relatives.
In its policy briefing to lawmakers, the ministry said it would discuss with North Korea issues related to holding the reunion event regularly, locating long-lost families and permitting people to visit their hometowns across the border, through Red Cross meetings.
But the lack of any concrete moves by Pyongyang on this goal has led some to doubt whether Kim will go along with the US's arrangements.
CJ Logistics said Tuesday it is transporting necessary material for this week's inter-Korean family reunions in North Korea.
"Even if he's gotten old and has his wrinkles on his face, there is still some of him left in his appearance", Kim Young-sun said after reuniting with his North Korean brother.
In his first visit to Israel since being appointed national security adviser Bolton has sought to highlight the flourishing US-Israel alliance, which he said "has never been stronger".
Chinese President Xi Jinping will visit Pyongyang on September 9 to attend celebrations of the 70th anniversary of North Korea, Singaporean media reported Saturday. Reports also say that Chinese power plants are lighting up Pyongyang at night, North Korean restaurants in China are reopening and an estimated 215 companies are jointly run by Chinese and North Koreans.
A fresh round of such editorializing over the weekend was taken as a signal that American officials have sternly confronted North Korea about suspected secret nuclear weapons facilities.