Egyptians have poured on to the streets, swelling crowds opposition leaders hoped would number into the millions to persuade Islamist President Mohammed Morsi to resign.

Waving national flags, tens of thousands gathered yesterday on Cairo's Tahrir Square, seat of the 2011 uprising against his predecessor Hosni Mubarak.

"The people want the fall of the regime" they chanted, this time not against an ageing dictator but against their first elected leader, who took office only a year ago.

As the working day ended, more joined them on the otherwise deserted streets of the capital.

Many are angry at Morsi's Muslim Brotherhood, saying it has hijacked the revolution through a series of electoral victories to monopolise power and push through Islamic law.

Others are simply frustrated by the economic crisis, deepened by political deadlock, over which Mr Morsi has presided.

In other cities, thousands of protesters also gathered.

Security sources said three Brotherhood offices were set on fire by demonstrators in towns in the Nile Delta, the latest assaults in more than a week of street violence in which hundreds have been hurt and several killed, including an American student.

More than 10,000 Morsi supporters also congregated in the capital, beside a mosque not far from the suburban presidential palace. Mr Morsi is working elsewhere, but liberal protest organisers staged a sit-in outside the palace yesterday.

Mr Morsi has repeated his determination to ride out what he sees as an undemocratic attack on his electoral legitimacy, but he also offered to revise the new, Islamist-inspired constitution, saying clauses on religious authority that fuelled liberal resentment were not his choice.

He made a similar offer last week, after the head of the army issued a strong call for politicians to compromise. The opposition dismissed it as too little too late. They hope Mr Morsi will resign in the face of large numbers on the streets.

Some also seem to believe the army might force the president's hand. In Cairo, demonstrators stopped to shake hands and take photographs with soldiers guarding key buildings.

While many Egyptians are angry at Mr Morsi over the economy, many others fear more turmoil will make life worse.

Mr Morsi and the Brotherhood can hope protests fizzle out like they have previously. If they do not, some form of compromise, possibly arbitrated by the army, may be on the cards. Both sides insist they plan no violence, but accuse the other – and agents provocateurs from the old regime – of planning it. The US-equipped army, though showing little sign of wanting power, warns it may step in if deadlocked politicians let violence slip out of control.

US President Barack Obama called for dialogue and warned trouble in the biggest Arab nation could unsettle an already turbulent Middle East. Washington has evacuated non-essential personnel and reinforced security at its diplomatic missions.

Mr Morsi dismissed the demands to resign. If that became the norm, he said, "well, there will be people or opponents opposing the new president too, and a week or a month later they will ask him to step down".