Trump had said on Friday that he was "ready to restart US mediation" to resolve the question of water sharing along the Nile, upon which Egypt relies for 97 percent of its water needs.
Sisi said he welcomed "Trump's attention to the central importance of the Nile River issue for Egypt, as it represents the lifeline of the Egyptian people" in a post on X.
He added that he had a desire to cooperate with other countries along the river "without harming any party".
Inaugurated in September, the Grand Ethiopian Renaissance Dam (GERD) is the largest in Africa, at 1.8 kilometres (1.1 miles) long and 145 metres (475 feet) tall.
It is expected to double electricity production in Ethiopia -- the continent's second most populous country, where nearly half of the people lack power -- but Egypt fears it will throttle water supplies downstream.
The GERD is situated on western Ethiopia's Blue Nile, which later joins the White Nile in Sudan to form the Nile River, which flows through Egypt to the Mediterranean.
Sudan's army chief and de facto leader Abdel Fattah al-Burhan also welcomed Trump's mediation in the hope of "finding lasting and satisfying solutions that preserve the rights of all", he said on X Saturday.
Various efforts at mediation between the three countries over the last decade -- led at different times by the US, the World Bank, Russia and the United Arab Emirates -- have consistently failed.
Trump said in a letter to Sisi released by the White House on Friday that he hoped for a formula that would provide a predictable water supply for Egypt and Sudan, and let Ethiopia either sell or give electricity to the two downstream countries.
Sisi has called the dam an existential threat, while Ethiopia sees it as a national project of historic scale and a rare unifying symbol.
Trump had stunned both Egypt and Ethiopia near the end of his first term when he expressed hope for a resolution to the GERD question, saying that if not, Egypt would "blow up that dam".
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