Zuma, whose presidency has been marred by graft scandals and economic decline, has been in a weakened position since he was replaced as leader of the ANC by Deputy President Cyril Ramaphosa in December.
Earlier Tuesday, ANC leaders also called a special meeting of its decision-making National Executive Committee (NEC) for Wednesday in Cape Town, heralding a bid to unseat the besieged president.
Facing a no-confidence motion in parliament set for Feb. 22, Zuma has survived several attempts to oust him in the past. But this time around a significant part of the ANC wants him to step down well before his second term ends mid next year.
ANC Deputy Secretary-General Jessie Duarte told a news conference that senior party officials who met on Monday would take a proposal to the NEC for discussion on Wednesday, but she declined to disclose the proposal.
"The NWC (National Working Committee) has discussed the issue surrounding the future of President Zuma and the matter that we had discussed will be taken to the NEC," Duarte said, referring to Monday's meeting.
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."
Analysts said the proposal was likely to be on Zuma's future.
"A vote of no-confidence is not desirable, under any circumstances. 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," Duarte said.
"We will tell you exactly what the NEC has decided post our meeting."
The speaker of parliament, Baleka Mbete, postponed Zuma's speech due on Thursday. She said she met with Zuma, who was already writing to parliament to ask for the postponement of his address.
"A new date for the state of the nation address will be announced very soon," Mbete said.
The opposition parties had demanded that the speech be postponed until Zuma was removed from the leadership.
The rand, which has tended to strengthen on signs that Zuma could step down before his second term as president ends next year, was firmer on Tuesday.
The ANC's top six most powerful officials met Zuma late on Sunday at his official residence in Pretoria but there was no announcement of the outcome. Analysts said the senior officials had met Zuma to ask him to step down.
Duarte and party secretary-general and top-six member Ace Magashule have backed Zuma.
Some within the ANC and the opposition have said the Gupta family, friends of Zuma, have used their links with the president to win work with the state. The Guptas and Zuma have denied any wrongdoing.
Zuma chairs cabinet meetings
Zuma meanwhile was chairing routine cabinet committee meetings on Tuesday and not holding a "special cabinet meeting" as reported in local media, his spokesman said, as pressure mounted on the scandal-plagued leader to step down.
Bongani Ngqulunga said a full cabinet meeting was scheduled for Feb. 14, dismissing speculation in domestic media the embattled president had called a meeting to discuss his future with his cabinet colleagues.
"It's a routine meeting of cabinet committees, there is no special cabinet meeting going on," said Ngqulunga.
The influential Nelson Mandela Foundation said in a statement "time is of the essence - Zuma must go". Leader of the official opposition and head of the Democratic Alliance party Mmusi Maimane said in a statement: "We need a new beginning."
On Monday, Zuma met Zulu King Goodwill Zwelithini, the influential traditional head of South Africa's biggest ethnic group in the president's home province of KwaZulu-Natal.
A Zulu royal house insider told the BusinessDay newspaper that Zuma had refused a request from the Zulu king on Monday that he resign. Zuma had declined to resign saying that "if he resigns now it would mean that he would admitting that he had done something wrong", the insider told the newspaper.
(FRANCE 24 with REUTERS and AFP)a