Full relations would give the Catholic Church a legal framework to look after all of China's estimated 12 million Catholics and move on to focus on Catholic growth in a country where Protestant churches are already growing fast.
Catholics in China are split between those in "underground" communities that recognize the pope and those belonging to the state-controlled Catholic Patriotic Association, in which bishops are appointed by the government in collaboration with local church communities.
Under the formal deal, the Vatican would have a say in negotiations for the appointment of future bishops, the person told Reuters, declining to give details.
"It is not a great agreement, but we don't know what the situation will be like in 10 or 20 years," the person said. "It could even be worse."
"Afterwards we will still be like a bird in a cage, but the cage will be bigger," he said. "It is not easy. Suffering will continue. We will have to fight for every centimeter to increase the size of the cage."
The person rejected recent accusations by a senior cardinal that the Vatican was prepared to "sell out" the church in China, as well as media suggestions that Pope Francis was out of the loop on China negotiations.
He said the pope followed the China dossier very closely and had backed an offer made to two Chinese bishops loyal to the Vatican in which they would take on different positions in their dioceses to facilitate an overall accord with government-backed bishops.
Five of seven complex situations regarding "illegitimate bishops," those with government backing, had been resolved. They have asked for a pardon from the pope and to be made legitimate in the eyes of the church.
In December, with papal backing, a Vatican delegation went to China to make an offer relating to two Vatican-recognized bishops.
One, an 87-year-old prelate, would retire to make way for a state-backed bishop to succeed him. Under the scenario, the government would officially recognize the Vatican-backed prelate as "bishop emeritus."
Another Vatican-recognized bishop would become an auxiliary, or assistant, to one who had been appointed by the government. But even though he would effectively take on a lesser role, the government would grant him official recognition as part of the deal.
The person said both Vatican-backed prelates recognized they would be making sacrifices for the greater good of the church.
There had been what the person called "a gentleman's agreement" on seven government-backed bishops who would be made legitimate after seeking a papal pardon, but the person said that it still had to be formalized.
Dossiers on each making the case for legitimizing them have to be prepared for the pope.