The 45-year-old president has signalled his desire to remain in power until a successor is chosen in an election
"Direct negotiations are suspended", said Monsignor Marcel Utembi, the head of the Congolese bishops conference (CENCO).
The announcement ended 11th-hour talks on Saturday between Kabila's ruling party and opposition groups to find a path forward for the Democratic Republic of Congo after December 20, when Kabila's second term ends.
The political transition talks won't resume until December 21, Utembi said.
"There is no deal," said Jean-Marc Kabund, the secretary general for the Union for Democracy and Social Progress (UDPS) party of the main opposition leader, Etienne Tshisekedi.
"The ruling majority is sitting on its positions and refuses to offer any concessions on matters that require a political response," he said.
The Congolese constitution bans presidents from serving a third term, but Kabila has shown no signs of stepping down and no elections have yet been planned, prompting fears the situation could descend into violence.
The 45-year-old president has signalled his desire to remain in power until a successor is chosen in an election, but some opposition figures want him to hand over to a transitional leadership while awaiting the vote.
The CENCO-sponsored talks had brought together the ruling party and fringe opposition groups, allied against a mainstream opposition coalition led by the 84-year-old Tshisekedi.
The latter accuse Kabila of delaying the vote in the hope of tweaking the constitution to extend his family's hold over a nation hugely rich in minerals that is almost the size of western Europe.
Kabund, speaking to the media after the talks broke up, said: "We call on the people to carry out their responsibilities with respect to the... date marking the end of Kabila's mandate," referring to a midnight Tuesday deadline.
The church mediators have warned that failure to find a political settlement would lead to "an uncontrollable situation".
The international community has also warned the current tension could descend into violence, and Western diplomats in Kinshasa this week urged all non-essential nationals to leave the country.
The opposition coalition had threatened to mobilise in the streets from December 19 until Kabila left office, but did not immediately call for protests following the suspension of talks.
But on Saturday leaflets calling on people to remove Kabila from power began to circulate in Kinshasa, a city of 10 million where the head of state is notoriously unpopular.
Some two decades ago, Congo sunk into the deadliest conflict in modern African history, its two wars in the late 1990s and early 2000s dragging in at least six African armies and leaving more than three million dead.
A democratic handover would break ground for Congo's 70 million people who since independence from Belgium in 1960 have never witnessed political change at the ballot box.
In the last few years hundreds of people have died in political violence in the capital Kinshasa and elsewhere.
Tshisekedi and his allies had threatened nationwide protests from Monday to press Kabila to step down, but opted to hold off pending the negotiations.
The government has ordered that social networks including Facebook and WhatsApp be blocked from 11:59 pm (2259 GMT) on Sunday.
Police set up road checkpoints in Kinshasa, while the Republican Guard patrolled the capital's administrative district on Saturday near the presidential palace, AFP journalists reported.
A Belgian plane on a regular flight to dispatch material for military cooperation was sent back by authorities immediately after landing in Kinshasa on Saturday, a Western diplomat said.
The government frequently accuses Western powers of colluding with the opposition.