At least four people were killed and 200 injured during clashes Sunday between supporters and opponents of Morsi, Reuters said.
All four were killed in towns south of Cairo, while The Guardian said there has been no violence near the presidential palace yet.
In an exclusive interview with The Guardian, Morsi rejected calls to remove himself from office as well as requests for an early election.
"If we changed someone in office who [was elected] according to constitutional legitimacy – well, there will be people opposing the new president too, and a week or a month later they will ask him to step down," Morsi said.
The leader of Egypt's Islamist Nour party has urged Morsi to make some concessions to avert further bloodshed.
"We are worried about an escalation that will be hard to control, and that guns will have the loudest voice," Younis Makhyoun said. "The president is banking on the idea that this will be a mass protest, like the other protests, and there will be some losses, and then things will settle down. But this time matters are different."
The mass rally comes on the first anniversary of Morsi's election. Prior to Sunday, Morsi's opponents vowed mass protests and his supporters said they would hold counter rallies.
Under heavy guard, Morsi and his family were moved from their home in the presidential palace to a nearby military base for their own protection, Israel Radio reported. Officials fear violence may erupt and protesters may attempt to march on the presidential palace, the radio report said.
Thousands of protesters thronged to Tahrir Square and Sunday morning more protesters arrived at the site, Ahram Online said. Demonstrations were reported early Sunday in the Suez, Sharqia, Monofia and Gharbiya districts, the Egyptian website said. In Suez the army distributed flyers urging protesters to stay away from state institutions and prisons, the site said.
Morsi's opponents are demanding his resignation over his failed economic and security policies.
Opposition activists said more than 22 million people have signed a petition urging Morsi to step down.
Security was beefed up last week in preparation for the protests, with Egyptian soldiers and police deployed in Cairo and elsewhere in the country. Barbed wire was placed around government buildings and areas around Tahrir Square. Police checked cars and those entering shopping malls and other buildings were also subjected to security checks, Ahram Online said. Many banks, companies and public and private businesses instructed employees to take Sunday off for their own safety, the website said.
In the past week prior to Sunday's events, eight people have been killed and hundreds injured in clashes, Ynetnews.com said, including American college student Andrew Pochter, who was stabbed to death in Alexandria when protesters stormed a local Muslim Brotherhood office. Pochter had volunteered to teach English to children in the country, Ahram Online said.
In a televised speech to the nation last week, Morsi said, "I took responsibility for a country mired in corruption and was faced with a war to make me fail. Egypt's enemies have spared no effort to sabotage the democratic experience."
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