The reported decision to "recall" Zuma comes after marathon talks held by senior party officials that continued into the early hours of Tuesday.
If Zuma, 75, still did not budge, he would face a vote of confidence in Parliament that he was expected to lose, the BBC said.
He has been leader since 2009 but has been dogged by corruption allegations, which he has vehemently denied.
The ANC has not officially confirmed its plans, but party sources have described them to South African media outlets.
It is unclear how Zuma would respond to the formal request to step down, which was expected to be issued later on Tuesday.
Earlier, Ramaphosa left the ANC's national executive committee meeting to travel to tell Zuma that he would be recalled if he did not step down. He later returned to the ANC conclave.
In 2016, South Africa's highest court ruled that Zuma had violated the constitution when he failed to repay government money spent on his private home.
In 2017, the Supreme Court of Appeal ruled that he must face 18 counts of corruption, fraud, racketeering and money laundering relating to a 1999 arms deal.
Zuma's links to the wealthy India-born Gupta family, who are alleged to have influenced the government, have caused his popularity to plummet.
Both Zuma and the Guptas deny the allegations, the BBC said.
It would be very difficult though for him to resist a formal request to resign - known as a "recall", the report added.
However, he would then be expected to face a confidence vote in Parliament. The date for this has already been set as February 22.
Zuma has earlier survived other such votes, but he would not be expected to pull it off again.
South African media is calling President Zuma's seemingly inevitable exit "Zexit". His predecessor, Thabo Mbeki, resigned in 2008. He also had a power struggle with his deputy. That deputy was Jacob Zuma.
(This story has not been edited by Business Standard staff and is auto-generated from a syndicated feed.)