The ANC's highest decision-making body, the National Executive Council (NEC), will hold a meeting on Wednesday in which Zuma's presidency is expected to be discussed.
DA leader Mmusi Maimane said SONA could only proceed after Zuma is removed.
In a statement, the ANC said the meeting was called to discuss the "management of the transition" between the Zuma and Ramaphosa administrations.
Mr Zuma, in power since 2009, is fighting for survival.
Economic Freedom Fighters (EFF) secretary general Godrich Gardee.
Or who might do so if he were to resign unexpectedly.
ANC officials could not be reached for comment.
Zuma's delivery of the State of the Nation Address, originally schueduled for Thursday, has been postponed, parliament speaker Baleka Mbete announced.More news: Snow and ice weather warning for Northern Ireland
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Separately, Zuma's lawyers have submitted arguments to prosecutors about why he shouldn't be prosecuted for corruption charges tied to an arms deal two decades ago.
Some within the ANC and the opposition say the Gupta family, friends of Zuma, have used their links with the president to win work with the state.
While Zuma is due to step down in mid-2019, his nine-year tenure has been marred by a series of scandals and policy missteps.
Zuma met Cabinet ministers on Tuesday and, a day earlier, visited Zulu King Goodwill Zwelithini in his home base of KwaZulu-Natal province, which has been a strong base of support for the president.
"Our most important consideration is that we don't believe South Africa should wish for us to embarrass the president of the republic, in any way whatsoever", she said.
Sources told News24 that Zuma told the top six that he had done nothing wrong and was undeterred by threats of impeachment and a vote of no confidence set for February 22.
Unlike in August when Zuma survived a no-confidence vote, a significant portion of the ANC now wants him gone. She initially said she had no right to even consider the postponement - but has now signed off on exactly that. In recent years, opposition lawmakers have disrupted the speech, which traditionally.
The ANC's most powerful officials have already held private talks with President Zuma, but there's been no announcement of the outcome of the meeting.
Ralph Mathekga, who authored When Zuma Goes, said the postponement shows the mounting level of pressure on Zuma.