The development comes amid concerns that Egypt is slipping back into authoritarianism, eight years after a pro-democracy uprising ended autocratic President Hosni Mubarak's nearly three-decade rule. The amendments also boost the power of the military, already the dominant force in Egyptian politics.
Lawmakers are expected to vote on Thursday, after which the text of the amendments would be finalized by a special legislative committee and sent back to the assembly for a final decision within two months.
The 596-seat assembly, which is packed with el-Sissi's supporters, already gave its preliminary approval to the changes last week. The amendments are almost certain to be overwhelmingly approved by the legislature, but will also need to be put to a national referendum to become law.
The referendum is likely to take place before the Muslim holy month of Ramadan, which is expected to start in early May this year. Parliament Speaker Ali Abdel-Al opened Wednesday's session, telling lawmakers in the packed chamber that there will be a "national dialogue" and that "all opinions and trends will be included in the discussions."
Wednesday's parliament session was not broadcast on TV although by evening several videos emerged on social media showing lawmakers speaking for or against the amendments. Human Rights Watch said the amendments would undermine judicial independence and expand executive powers that are already being abused in Egypt.
"These amendments reinforce efforts of President el-Sissi's military-backed government to stifle people's ability to challenge those in power," said Michael Page of the New York-based group. "If the amendments are passed, there is a clear risk that they will formally give the armed forces unchecked authority."
The vote had initially been scheduled for next week, but was moved up. A coalition of nearly a dozen opposition parties has come out against the amendments, but on their own they will not be able to block them.
El-Sissi, who previously held the office of military chief, led the military's 2013 overthrow of the freely elected but divisive Islamist president, Mohammed Morsi, after protests against his rule. El-Sissi was elected president the following year and has since presided over an unprecedented crackdown on dissent. He was re-elected last year after all potentially serious challengers were jailed or pressured to exit the race.
Along with extending a president's term in office from four to six years, the amendments include a special article that applies only to el-Sissi and allows him to run for two more six-year terms after his current term expires in 2022.
The amendments also envisage the office of one or two vice presidents, a revived Senate, and a 25 percent quota for women in parliament. They call for "adequate" representation for workers, farmers, young people and people with special needs in the legislature.
The president would have the power to appoint top judges and bypass judiciary oversight in vetting draft legislation before it is voted into law. The amendments declare the country's military "guardian and protector" of the Egyptian state, democracy and the constitution, while also granting military courts wider jurisdiction in trying civilians.
In the last three years, over 15,000 civilians, including children, have been referred to military prosecution in Egypt, according to Human Rights Watch.