SEOUL—South Korea’s leader proposed changing the constitution to allow presidents to serve up to eight years over two terms, instead of the current limit of one five-year term, a shift that advocates say would provide more stability in Seoul’s policies toward North Korea.

President Moon Jae-in’s office on Thursday said officials were reviewing the proposal, which was among his campaign promises last year. Mr. Moon’s predecessor, Park Geun-hye, had also promised to push through a plan to extend term limits, but legislation...