Pope Francis has decided to accept the legitimacy of seven Catholic bishops appointed by the Chinese government, a concession that the Holy See hopes will lead Beijing to recognize his authority as head of the Catholic Church in China, according to a person familiar with the plan.

For years, the Vatican didn’t recognize the bishops’ ordinations, which were carried out in defiance of the pope and considered illicit, part of a long-running standoff between the Catholic Church and China’s officially atheist Communist Party....