On Monday and Tuesday, the former president’s supporters and unidentified opponents fought running battles with firearms, bottles and rocks near Tahrir Square in Cairo and on the edges of two protest sites that have been held by Mr. Morsi’s Islamist supporters for weeks. The confrontations have escalated as the Islamists have broadened their demonstrations, marching in cities across the country as part of a determined but so far fruitless effort to restore Mr. Morsi to power.

There has been little sign of a solution to the standoff, which started after the militaryremoved Mr. Morsi from power on July 3 after mass demonstrations throughout Egypt calling for his ouster. As Egypt’s new military-backed government has moved swiftly to assert its authority, Mr. Morsi’s movement, the Muslim Brotherhood, has refused, at least publicly, to abandon its positions.

On Monday, the Brotherhood released a statement reiterating its demands that the “coup must be halted” and that the president, the Constitution and the Parliament must be restored.

“It was the putschists’ intention to sow despair in the hearts of the Egyptian people,” the statement said.

The authorities appear to be unable or unwilling to halt the violence. The fighting near the university spread to streets that have become a virtual garrison for army troops and riot police officers. Mr. Morsi’s supporters have accused the police of joining the fight on the side of their opponents, and said that early Tuesday, police officers had fired on demonstrators in at least two locations.

That claim was impossible to verify. In front of Cairo University on Tuesday morning, cars damaged in the fighting were hauled away as protesters slept in tents. A man yelled “We only want security!” as friends tried to comfort him. A mother and her two children, carrying suitcases, walked away from the square, toward entrances once flimsily barricaded and now fortified with walls.