But a deal is not imminent, experts said, nor would it be universally welcomed within the Catholic Church. The latest developments have prompted one retired cardinal to make an emotional appeal to the pope on the bishops' behalf.
Beijing broke off diplomatic relations with the Vatican in 1951, not long after the Communist Party came to power. Since then, two parallel Catholic churches have grown up in China: the Chinese Catholic Patriotic Association (CCPA) run by state-appointed bishops and an underground church with many bishops appointed by the Vatican.
The Vatican is eager to bring the entire Chinese Catholic community of about 10 million to 12 million people back into the fold, and talks have been underway since 2014. Yet the question of whether the Chinese state or the Holy See has ultimate authority over the appointment of bishops has been a major sticking point.
In recent weeks, there were signs of a breakthrough, as first reported by the Catholic website AsiaNews.it.
Last month, a delegation from the Vatican traveled to China and met Bishop Peter Zhuang Jianjian, 88, who presides over the church in Shantou in the southern province of Guangdong.
In a meeting in Beijing, they asked Zhuang to retire in favor of a bishop appointed by the Chinese government, Huang Bingzhang, who is also a member of China's rubber-stamp parliament, the National People's Congress, and was excommunicated by the Vatican in 2011.
Zhuang was in tears at this demand, the website reported.
Another Vatican-appointed bishop, Joseph Guo Xijin in eastern Fujian province, was also asked to downgrade himself and become assistant to a CCPA bishop, Vincent Zhan Silu, the website reported.
On Monday, retired Hong Kong cardinal Joseph Zen confirmed the account reported by AsiaNews.it and said he had flown to Rome to meet the pope on behalf of Zhuang.
He said that he was granted a private audience Jan. 12 and that the pope promised to look into the matter.
Zen added that the pope had warned officials "not to create another Mindszenty case" — suggesting a certain degree of sympathy with the retired cardinal's point of view.
Cardinal Jozsef Mindszenty, head of the Catholic Church in Hungary, was imprisoned in the 1940s under communist rule, spent many years living at the U.S. Embassy and eventually was forced to leave the country to be replaced by someone acceptable to the communist government.
But the Vatican denied that there was a difference of opinion between the pope and his negotiating team.
"The pope is in constant contact with his collaborators, in particular in the Secretariat of State, on Chinese issues, and is informed by them faithfully and in detail on the situation of the Catholic Church in China and on the steps in the dialogue in progress between the Holy See and the People's Republic of China, which he follows with special attention," Greg Burke, the Vatican's media office director, said in a statement.
"It is therefore surprising and regrettable that the contrary is affirmed by people in the Church, thus fostering confusion and controversy," the statement added.
Zen has become the most vocal opponent of rapprochement between the Vatican and the Communist Party, and he reiterated those concerns in a blog post Monday.
"Is it not good to try to find mutual ground to bridge the decades-long divide between the Vatican and China?" he asked. "But can there be anything really 'mutual' with a totalitarian regime? Either you surrender or you accept persecution, but remaining faithful to yourself."
Zen compared it to making a deal between Saint Joseph, husband of Jesus' mother, Mary, and King Herod, who in the Bible ordered the execution of young male children.
"So, do I think that the Vatican is selling out the Catholic Church in China? Yes, definitely, if they go in the direction which is obvious from all what they are doing in recent years and months," Zen wrote.
But Francesco Sisci, a senior researcher at the Center of European Studies at the Renmin University of China and a Vatican affairs expert, said he does not think Zen represents mainstream opinion within the Catholic Church.
The State Department classifies China as a "country of particular concern" for having engaged in or tolerated particularly severe violations of religious freedom. Human rights groups say the Muslim Uighur minority, Tibetan Buddhists and members of the Falun Gong spiritual movement often face more systematic problems than Christians, although severe repression of unauthorized "house" churches has been reported.
In 2014, more than 2,000 Christian buildings and 600 crosses were demolished in a campaign against "illegal" buildings in the eastern province of Zhejiang, the State Department said, and a local pastor and his wife were imprisoned for 14 and 12 years, respectively, for opposing the campaign.
Nevertheless, the number of practicing Christians is rising rapidly in China and could total about 100 million people by some estimates.
Sisci said that thousands of Catholic missionaries work in China, with the full knowledge of authorities, and that expulsions are "extremely rare."
Nevertheless, he said it was a sensitive time for the Vatican to make major concessions to China.
"The Holy See is torn because of the papacy's stand on moral issues, on social issues. So the China issue, which is extremely controversial, can become a fuse that can rip the church apart," he warned.
But Sisci took heart from an article in China's state-run Global Times newspaper this week that appeared to acknowledge the Vatican's difficulties in reaching a deal.
"Never before have they realized the internal difficulties of the Holy See," he said, perhaps implying more flexibility from China. "This in itself is a breakthrough."
The Vatican is one of the few states that maintains diplomatic relations with Taiwan, and many people there fear it would be forced to break those ties if there is rapprochement with Beijing. But Sisci said this was premature, saying that the Vatican was not seeking full diplomatic ties with Beijing or anything that would damage Taiwan's interests.