The €3.3 billion ($4 billion) Grand Renaissance Dam, the largest hydroelectric project in Africa, is being built on the Blue Nile tributary of the river, which is Egypt's main source of water for consumption, industry and farming, Deutsche Welle wrote.
Egypt has long worried that Ethiopia's planned Grand Renaissance Dam would reduce the flow of freshwater from the Nile to its territory.
At Sunday's meeting in Cairo, Ethiopian Prime Minister Abiy Ahmed reassured President Abdel-Fattah al-Sissi that his country had "no desire to harm the Egyptian people."
"We will take care of the Nile, and we will preserve your share of the Nile’s water and we will work to increase this share," the Ethiopian prime minister promised.
The dam is being built near the Ethiopia-Sudan border and is already 63 percent complete.
Addis Ababa hopes the finished project can transform it into one of Africa’s key energy hubs.
In May, the irrigation ministers of all three countries that share the Nile agreed to set up a study group on how the dam would be filled and meet once every six months to discuss the situation.