Ramaphosa said at a party rally in Cape Town on Sunday he wanted to replace "a period of difficulty, disunity and discord" with "a new beginning" for the party.
"We know you want this matter to be finalised," he said to rapturous cheering, vowing to tackle the corruption that has tarnished Zuma's government.
Zuma has clung to power after rejecting a request by his party's senior officials to resign a week ago.
The powerful committee could recall the president from office, though he would be under no constitutional obligation to obey the order.
"We know you want closure -- we will be doing so keeping our eyes on what is in the interests of all our people," Ramaphosa said to loud applause on Sunday.
"The National Executive Committee of the ANC will be meeting tomorrow to discuss this very matter -- and because our people want this matter to be finalised, the NEC will be doing precisely that."
Litha Madita, 48, an NGO worker from Cape Town, welcomed the announcement of the NEC meeting, adding that Ramaphosa has spoken "to the aspirations of the South Africans".
"It brings hope that there is a new venture we are getting into.
"But we have to respect (Zuma) as a former president of the ANC. It is important not to disrupt the country or bring violence into the country."
Zuma's presidency has been marred by corruption scandals, slow economic growth and record unemployment that have fuelled public anger.
The stalemate over Zuma's departure has left Africa's most developed economy in limbo, with a series of public events cancelled last week including Thursday's State of the Nation address to parliament.
Opposition parties last week had threatened a "national shutdown" in response to Zuma's refusal to resign -- although it was unclear if the action would go ahead.
Dispute over exit deal?
Zuma's hold over the ANC was shaken in December when his chosen successor -- his former wife Nkosazana Dlamini-Zuma -- narrowly lost out to Ramaphosa in a vote to be the new party leader.
Sunday's rally was part of ANC celebrations marking 100 years since late leader Nelson Mandela's birth -- as well as efforts by Ramaphosa to try to revive the party's tainted reputation ahead of next year's general election.
"We have arrived at a moment in the history of our country where we can relive that moment when Nelson Mandela was released... we have a new mood right across the country, we can capture that mood and move forward," said Ramaphosa.
He was speaking to mark the 28th anniversary of the speech Mandela gave in the same location after being released from prison.
It is understood that a key sticking point in the negotiations is the potentially ruinous legal fees Zuma is facing from prolonged court battles against multiple criminal cases.
He is also reportedly seeking legal protection for his family and other associates who have been involved in controversial deals.
"Even if the ANC meeting on Monday decides Zuma needs to step down, he can still refuse because they have no legal authority,"Mcebisi Ndletyana, politics professor at the University of Johannesburg, told AFP.
"He is not willing to step down voluntarily. They need to close this thing early this week."
Opposition parties are calling for a parliamentary vote of no-confidence within days.
Decision time?
The ANC has insisted there will be no delay to the budget, which is due on February 21.
Zuma has not spoken publicly since being asked to resign by senior ANC officials on February 4.
In 2008, the party pushed out then-president Thabo Mbeki over allegations of abuse of power.
Under Zuma, the ANC won less than 54 percent of the vote in local elections in 2016.
That was its worst electoral performance since coming to power with Mandela at the helm in 1994.
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