Ramaphosa replaced the president as leader of the ruling African National Congress in December.
Two South African opposition parties on Thursday urged Parliament Speaker Baleka Mbete to schedule an urgent sitting of Parliament for the motion of no confidence in President Jacob Zuma to be debated and voted on.
He said the pair aim to conclude talks on the president's future within days.
The divisions Zuma's leadership has exposed within the ANC were evident outside the party's headquarters in Johannesburg on Monday (Tuesday NZT), where the president's supporters and opponents staged rival protests amid a strong police presence.
She added that on Wednesday the NEC "will be discussing a matter of serious concern to all of us within the ANC, and of course a matter of great public interest to the people of South Africa".
Zuma is still fighting almost 800 counts of corruption over an arms deal from the late 1990s and his ties with the Gupta brothers, a family of wealthy Indian-born businessmen, are the subject of a judicial inquiry on grand-level corruption.
They've started calling it "Zexit" here - the long, fractious process of nudging President Jacob Zuma out of his job.
ANC secretary-general Ace Magashule briefs the media after an ANC parliamentary caucus meeting on 8 February 2018.
Senior ANC official Paul Mashatile, who along with 5 others, on Sunday met with Zuma to convince him to resign remarked "Zuma basically said to us: 'I'm not going anywhere".
That appeared to be borne out at the 2016 local elections, when the ANC lost ground to the opposition Democratic Alliance (DA) and Economic Freedom Fighters (EFF).
South Africa had seen "systematic looting" under Mr Zuma's rule, and he "must go sooner rather than later", the foundation said.
Zuma's spokesman Bongani Ngqulunga said the meeting with Zwelithini was "a longstanding courtesy meeting between the President and His Majesty which was initially meant to take place in January but was postponed due to diary challenges on both sides".
Independent political analyst Ralph Mathekga told AFP that Mr Zuma's departure had been mishandled by the ANC.
As a result of his stance, the ruling party's NEC looked to have little choice but to officially recall him from office, which would have been a major embarrassment for a man who devoted his life to the former liberation movement.
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