North Korean leader Kim Jong Un has invited Pope Francis to Pyongyang, South Korea’s presidential office said Tuesday, in a new diplomatic outreach by the repressive regime as it negotiates a nuclear-disarmament deal with Washington.

A papal visit to Pyongyang could risk criticism of the Vatican, which recently signed a controversial agreement with China on the appointment of bishops in the world’s most populous country. North Korea has among the world’s worst human-rights records and operates a network of political prison...