Initial tallies reported by state media gave 90 per cent support for the new charter, which the military-installed authorities say provides greater protection for freedom of speech and women's rights.
The result of the referendum was always a foregone conclusion with the Muslim Brotherhood, designated a "terrorist organisation" by the government, calling for a boycott and little sign of a "No" campaign in the run-up to the vote.
Sisi has said he is prepared to run if there is enough popular support and the two-day referendum that wrapped up yesterday provides the first concrete test.
The military's spokesman thanked the "masses" of voters for taking part in what he called the "heroic battle of the referendum".
More From This Section
Egyptian state-run dailies hailed the vote.
"The people say 'Yes'," said a front-page headline in Al-Akhbar, while Al-Ahram reported that 90 per cent of voters had voted in favour of the charter.
At least 444 people were arrested for protesting and disrupting polling over the two-day vote, the interior ministry said.
The government said it was aiming for a larger turnout than the 33 per cent of the country's 53 million registered voters who cast a ballot in a constitutional referendum under Morsi in 2012, with 64 per cent voting yes.
"We are hoping it exceeds 50 per cent," government spokesman Hany Salah told AFP.
The new text has done away with much of the Islamist-inspired wording of Morsi's charter. It bolsters the military's powers and allows it to try civilians for attacks on the armed forces.
US Secretary of State John Kerry said he hoped the referendum would be "transparent and accountable".
"But we don't know yet," he told reporters in Kuwait.
On Tuesday, the State Department said a bill Congress is expected to pass tomorrow will allow the White House to unfreeze all USD 1.5 billion in US aid if it can certify Egypt "has held a constitutional referendum, and is taking steps to support a democratic transition".