VATICAN CITY—Negotiators for the Vatican and Beijing reached a compromise on who selects Catholic bishops in China, said people familiar with the matter, potentially marking a major step toward ending six decades of estrangement.

If Pope Francis and Chinese leaders sign off on the proposed deal, the pope would accept eight bishops ordained by the Chinese government without the Vatican’s permission. But the deal would leave many other issues unresolved, including the role of China’s state-run Catholic institutions.

Negotiators...