The Prime Minister's Office of Ethiopia said the three countries agreed that the Nile and the Gerd "are African issues that must be given African solutions".
On Friday, the three countries agreed to form a committee of legal and technical experts to draft a final binding deal, and to "refrain from taking any unilateral measures, including the filling of the dam, before the agreement is reached", the Egyptian presidency said in a statement. Ethiopia did not comment explicitly on the start of the filling period.
Ethiopia has banked its hopes of development on the GERD and has said it has the potential to lift millions of Ethiopians out of poverty.
Years of negotiations have left the three countries unable to agree on how Ethiopia will operate the Grand Ethiopian Renaissance Dam (GERD) and fill its reservoir while also protecting Egypt's water source.
African Union Commission Chairman Moussa Faki Mahamat said the countries "agreed to an AU-led process to resolve outstanding issues", without elaborating.
Ethiopia says it will take three to seven years to fill the dam reservoir with 4.9 billion cubic meters of water for generation of electricity, while Egypt says it will affect the natural flow of the Blue Nile that provides it with 87 per cent of its water and has been pushing for gradual filling of between seven and 10 years to guarantee its annual share.
The Grand Ethiopian Renaissance Dam (GERD), set to be Africa's largest hydroelectric project, has been a source of tension in the Nile basin ever since Ethiopia broke ground on it almost a decade ago.
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According to the statement, Ramaphosa expressed his aspiration to intensify coordination between the two countries during the coming period, added Radi, noting that the South African President praised the honest and constructive political will that Egypt has always shown to reach a solution to the dam crisis.
"To ensure successful negotiations, all parties shall pledge to not make any unilateral decisions, as well as halt the dam's filling until reaching an agreement preserving the three countries' interests", Sisi said.
Sources within the UN told The EastAfrican that the issue has divided the five permanent members of the UN Security Council; the US, France, Russia, the United Kingdom and China.
Ethiopia criticised Egypt for detailing its grievances over the dam in a May letter to the UN Security Council - a move it described as a bad faith attempt to "exert external diplomatic pressure".
Egypt last week called on the United Nations Security Council to intervene to restart talks on the mega-project and warned that filling the dam without a deal would "threaten worldwide peace and security".
The Egypt-Ethiopia dispute now raises the question as to why most Nile Basin countries have not ratified the CFA in order to make the agreement take effect. Construction of the Dam started in April 2011.