China and the Vatican are set to sign a landmark agreement later this month ending a long struggle between Beijing’s Communist rulers and the pope over who chooses the leaders of Catholicism in the world’s most populous country, according to two people familiar with the matter.

Reactions to the deal, which gives both sides a say in appointing the church’s bishops in China, are likely to be sharply divided, with some hailing a diplomatic coup by the Vatican that draws China closer to the West and others warning of an important...