If the deal does succeed, it would be a major achievement for the Catholic church, which has been mediating talks in an attempt to prevent Democratic Republic of Congo sliding back into years of anarchy and civil war.
Pope Francis has heaped pressured on Kabila and the opposition to find a peaceful solution to the crisis in Congo.
"At first glance, a miracle is possible and the bishops have won their bet," Albert Moleka, former chief of staff to the leader of the main opposition bloc Etienne Tshisekedi, said.
A government spokesman said the proposal would be presented to the full delegation at the talks on Friday afternoon, but he declined to comment on the specifics of the deal.
In return for Kabila staying on for another year, the constitution will not be changed to let him stand for a third term, a prime minister will be named from the main opposition bloc and Tshisekedi will oversee the deal's implementation, opposition leaders Martin Fayulu and Jose Endundo told Reuters.
"Kabila stays for one year," Fayulu said. "He will not try to stand for a new term."
However, Kabila himself has so far said nothing and the parties have yet to sign the deal, which requires final approval by all delegates at the talks.
It was not immediately clear how the wider population would react to an agreement. On Twitter, a leader of the youth activist group Filimbi, Floribert Anzuluni, said it constituted "high treason by everyone".
However, many in the capital Kinshasa, who spoke to Reuters on Thursday after the protests subsided, said they were tired of the violence and hoped for a negotiated settlement.
Church leaders have presented these talks as a last ditch effort to prevent violence spinning out of control after a bloody week that saw protesters killed and deadly clashes between various ethnic militia across the country.
KABILLA'S TENURE EXPIRED TUESDAY
Kabila's mandate expired on Tuesday but he has remained in office as a presidential election scheduled for last month was postponed until at least April 2018 because of what the government said were delays registering voters.
He has declined to commit publicly to not changing the constitution, leading to accusations that he is trying to cling to power.
Kabila's allies have repeatedly argued that he is committed to respecting the constitution but that promising to step down would make him a lame duck and possibly spark a power struggle that could put his life in danger.
His father, President Laurent Kabila, was assassinated by a bodyguard in 2001 and Congo has never had a peaceful transition of power.
This week's protests, which killed at least 34 people, and a disparate clashes between ethnic militias have raised fears Congo is heading towards another major armed conflict. Millions were killed in wars between 1996 and 2003.
BLOODY PROTESTS
The head of the U.N. human rights agency said on Friday that Congolese security forces had killed at least 40 people and arrested 460 in protests this week.
The violence has raised fears Congo is heading toward another major armed conflict, a risk that has prompted several donor nations to condemn Kabila for failing to stand down.
Millions were killed in wars between 1996 and 2003.
"Most of ... (Kabila's coalition) would welcome this (deal) because they’re under so much pressure," said Pascal Kambale, a Congolese human rights lawyer working for the Open Society Foundations.
But Jean Marc Kabund, the secretary general of Congo's largest opposition party, the UDPS, warned that the deal was not yet a sure thing.
"Today is the last day (of negotiations)," he told Reuters. "It's make it or break it."
It was not immediately clear how the wider population would react to an agreement. On Twitter, a leader of the youth activist group Filimbi, Floribert Anzuluni, said it constituted "high treason by everyone".
However, many in the capital Kinshasa, who spoke to Reuters after the protests subsided, said they were tired of the violence and hoped for a negotiated settlement.