SEOUL—North Korean leader Kim Jong Un invited his South Korean counterpart for a summit in Pyongyang, presenting Seoul with a quandary: accept, and risk heightening differences with the U.S. over how to rein in the North’s nuclear program, or decline, and risk a chance to build on a thaw in cross-border relations.

The invitation was relayed by Kim Yo Jong, the North’s special envoy and sister of dictator Kim Jong Un, to South Korean President Moon Jae-in during a three-hour meeting at the presidential Blue House in Seoul on...