Deaths were reported in Cairo, the Assiut district, Kafr el-Sheik and other areas, al-Masry al-Youm reported.
On the second day of the protests, Tamarod, the rebel campaign that helped organize them, said if Morsi doesn't leave office by Tuesday evening, a civil disobedience campaign will include a march on the presidential place, Ahram Online said.
"In the name of 22 million Egyptians [who signed a petition] we now announce that Mohamed Morsi is no longer a legitimate president of the Arab Republic of Egypt," a statement issued by the group said.
The scale of the demonstrations, a year after crowds in Cairo's Tahrir Square cheered Morsi's inauguration, appeared to exceed the massive street protests that led to the ouster of President Hosni Mubarak in 2011, The New York Times reported.
Tens of thousands of people were in Tahrir Square Monday, the BBC reported. Millions protested in cities across the country Sunday.
Other demonstrations persisted outside the presidential palace in the suburb of Heliopolis several northeast of Cairo.
Thousands of Muslim Brotherhood supporters of Morsi staged a separate protest near the palace to defend what they called the president's "legitimacy" and to defend the palace if protesters tried to attack.
Many brought batons, pipes, bats, hard hats or motorcycle helmets, the Times said.
"We will sacrifice our lives for our religion," some chanted. "Morsi's men are everywhere."
Protesters ransacked Muslim Brotherhood offices around the country, the Times said.
In Cairo, hundreds set fire to the nearly empty Brotherhood headquarters, throwing stones, Molotov cocktails and fireworks for hours.
Brotherhood members inside fired birdshot at the attackers, wounding several, the Times said.
Police and security forces did nothing to stop the assault or the arson, the newspaper said.
Police, left over and still intact from the Mubarak government, are in open revolt against Morsi, the Times said.
Several officers in uniform were seen among the Tahrir Square protesters calling for Morsi's ouster and asking the military to intervene.
"There are protests -- this is a reality," Morsi spokesman Omar Amer said at a midnight news conference.
"We don't underestimate the scale of the protests, and we don't underestimate the scale of the demands," he said.
Amer said the administration was open to discussing any demands consistent with the Egyptian Constitution.
The constitution and the manner in which it was adopted have been a focus of anti-Morsi demonstrations since last year.
When journalists asked questions of Amer, he appeared to become exasperated, returning with questions of his own.
"Do you have a better idea? Do you have an initiative?" he said. "Suggest a solution and we're willing to consider it seriously."