El-Sissi led the military coup against former Islamist president Mohammed Morsi last July and hinted as recently as Saturday, while speaking at an army seminar in Cairo, that he would run for president "at the request of the people."
The results of the constitutional referendum "are one of the most important indicators" of whether or not el-Sissi will launch a bid for the presidency, said former army general Sameh Seif al-Yazal.
Egypt's government and its supporters have portrayed the vote as a choice between stability and "terrorism," after officially declaring the Brotherhood a terrorist organization and criminalizing its activities last month. In recent weeks, thousands of Brotherhood supporters and non-Islamist activists opposed to military rule have been rounded up and jailed on questionable charges, including some under a draconian new law that severely restricts demonstrations.
El-Sissi spearheaded the crackdown on Brotherhood loyalists, which has given rise to sporadic but deadly militant attacks on security forces.
The pro-Morsi Anti-Coup Alliance called Monday for Egyptians to boycott what they called a "sham referendum" on "the military's illegitimate constitution."
Monday night, police arrested members of the Islamist-leaning Strong Egypt Party for publicly advocating a vote against the charter. At least one of the detainees will be charged under the section of the penal code that deals with crimes of terrorism, according to Human Rights Watch.
Across the capital Tuesday, posters urged voters to "Vote 'yes' to the constitution and 'no' to terrorism."
Of the 11 deaths confirmed by officials, four were in Sohag province, in Egypt's far south, and the others came in clashes in Giza and Beni Suef, all outside Cairo.
"I'm voting yes because I have respect for my country," said Sayed Ali, 42, a newspaper vendor who witnessed a bombing outside a courthouse in Giza that caused damage but no casualties. "Because there is a terrorist organization that is trying to destroy us."
The stark political polarization and heightened rhetoric in Egypt have clouded the debate over the new charter's content, which shields the military from civilian oversight and grants the judiciary near-autonomy, experts say.
Rights groups have denounced the document for leaving articles on civil rights and liberties open to interpretation by legislative bodies.
The constitution "does not provide for a political system that will guarantee and protect the rights and liberties it ostensibly recognizes from violations through legislation and attacks by the security apparatus," the Cairo Institute for Human Rights Studies said in a statement Sunday.
But others have praised it for breaking with the Islamist-tinted 2012 constitution put forward by Morsi that critics said gave religion and religious bodies an outsize role in political life.
The charter "has reclaimed its identity as a civil constitution," Essam el-Islamboli, a lawyer and constitutional expert, said of the draft document. "It is free from the articles that tried to turn us into an Islamic emirate and has reaffirmed Egypt's national identity."
Many voters in Cairo also saw their yes votes as repudiation of the political Islam of the Muslim Brotherhood. Passing the constitution, they said, would move the country toward stability.
"I voted for Morsi, but this was my mistake. So now I am voting yes for this constitution," said Outeloloub Hassan, a 75-year-old woman wearing a niqab, or full face veil.
"It is also a vote for el-Sissi," Hassan said of her ballot. "And Egypt will be a safe country again."
Lara El-Gibaly contributed to this report.