Though expectations have been building, a breakthrough has been elusive given the Communist government’s deep suspicion of foreign and religious influences as subversive, and the fears of Chinese Catholics wary of state interference in their faith, said people closely following the talks. “There are still difficult issues that are not yet agreed upon,” said the Rev. Jeroom Heyndrickx, the acting director of the Ferdinand Verbiest Institute in Belgium, which studies Catholicism in China.

The central dispute is over the power to name new bishops and the fate of existing bishops in China. For the Catholic Church, bishops are divine successors of the apostles, to be appointed by the pope. But China has long insisted on controlling ordinations, arguing that anything else amounts to interference in its internal affairs.

Most Chinese bishops are recognized by both the Vatican and the Chinese authorities, but there are several in the state-backed church who are excommunicated and working without papal approval, including some rumored to have broken their vows of chastity and fathered children.

There are also more than two dozen underground bishops, many of whom are viewed with suspicion by the government and a few of whom are believed to be in prison.

Any deal would have to decide what happens to both groups. “The Vatican can’t be seen as selling out people who have suffered and gone to jail for their faith,” said a Vatican official, who requested anonymity to speak candidly about the secretive talks.

There is little doubt of Francis’ enthusiasm for China.

In 2014, he sent a greeting to President Xi while flying through Chinese airspace on his way to South Korea. And his encounter with Joseph Xu Honggen, the bishop of Suzhou, in St. Peter’s Square last month was said to be the first public meeting between a pope and a bishop resident in mainland China since the Communist Revolution.