Some Congolese have expressed doubt that the first election results will be released on Sunday as expected, as observers have said the internet and text messaging outage has slowed the transmission of election information.
"The announcement of voting trends by Priest Nshole is likely to brainwash the population while preparing an insurrection that CENCO alone will be responsible for", commission president Corneille Nangaa wrote in a letter to CENCO president Marcel Utembi, seen by Reuters.
A new date has not been announced.
Is the December 30 election in the Democratic Republic of Congo going down the 2011 way?
The DRC Bishops Conference (CENCO), said on 3 January that "data in its possession from vote counting reports from polling stations designates the selection of one candidate as president". Because Congo's regulations say only the electoral commission can announce election results, the church did not name the victor.
Electoral officials in the Democratic Republic of Congo have postponed the release of the presidential election results despite enormous calls for the outcome to be published.
CENCO deployed 40,000 observers nationwide to monitor the presidential and parliamentary polls.
"It said despite issued afflicting the process: "... millions of Congolese citizens turned out peacefully across DRC on election day to cast their ballots and make their voices heard.
Congo's ruling party, which backs Kabila's preferred candidate, Emmanuel Ramazani Shadary, called the church's statement "irresponsible and anarchist".
The government blocked Internet access and SMS services across the country, which the opposition claimed was proof of their fears that the results were being manipulated.
Opposition candidates Martin Fayulu and Felix Tshisekedi have accused the government of widespread irregularities, however.
DR Congo's powerful Catholic Church, which deployed more than 40,000 observers to monitor the elections, on Thursday said it knew who had won the vote but did not reveal who it was.
Any dispute over the results could further destabilise Congo's volatile eastern provinces, where wars around the turn of the century resulted in millions of deaths, most from hunger and disease, and where dozens of militia remain active.
"Tensions were mounting while the CENI tabulated the results, notably in light of posturing by parties and candidates", Leila Zerrougui, head of the UN Stabilization Mission in Democratic Republic of Congo, told the meeting, according to the internal report.
Sunday's vote, preceded by repeated delays, was relatively peaceful.
Washington has also urged Kinshasa to release "accurate" results and warned of sanctions against anyone seeking to "undermine the democratic process" in the former Belgian colony. According to the report, only the electoral commission is allowed to announce the victor.
While Congo has been largely calm on and after the December 30 vote, Trump's letter to House Speaker Nancy Pelosi said about 80 military personnel and "appropriate combat equipment" had deployed to nearby Gabon to support the security of US citizens and staffers and diplomatic facilities.