Tensions have run high since the president announced a controversial decree in late November expanding his decision-making powers beyond judicial review. Morsy has since partially dropped the decree, but opponents remain on edge ahead of a Saturday referendum on a divisive draft constitution.
Evidence of the tensions surrounding the vote could be seen in measures that the government has announced days before the vote.
On Tuesday, Morsy amended a law so that voters cannot cast their ballots outside their electoral districts, as they had in the past. Being able to vote anywhere had been a convenience, a presidential statement said, but it creates a burden on electoral officials.
The purpose of limiting voting to one's own district avoids "concerns about the fairness of the electoral process," the statement said.
Earlier, the government granted the military the power to make arrests during the electoral period, a power previously limited to police.
The move is designed to secure the voting process and will be rolled back once the election results are published, presidential spokesman Yasser Ali said.
The government is also allowing Egyptians living abroad to vote at 150 Egyptian embassies and 11 consulates worldwide, officials said. As many as 586,000 expats are registered to vote in the referendum.
Islamists formulated and voted to approve the national charter before handing it off to the general public to vote on, as representatives of other political and religious backgrounds quit the process in protest.
Before the separate but competing rallies Tuesday, attackers injured nine protesters with bird shot pellets in an assault on Tahrir Square in central Cairo before dawn, a Health Ministry official said.
Four were hospitalized with critical injuries, Dr. Mohamed Sultan said.
Dozens of assailants stormed the roundabout from three directions at 1:30 a.m., throwing Molotov cocktails and firing bird shot at protesters, said Mohamed Harbia, an activist who spent the night at the square.
Two protesters were wounded in the chest and one in the groin, said Harbia, who complained that ambulances took half an hour to arrive.