By Shaimaa Fayed and Yasmine Saleh

CAIRO, June 30 (Reuters) - Egyptians poured onto the streets on Sunday, swelling crowds that opposition leaders hope will number into the millions by evening and persuade Islamist President Mohamed Mursi to resign.

Waving national flags, tens of thousands gathered 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 ever elected leader, who took office only a year ago to the day.

As the working day ended and the heat of the sun eased, more joined them on the otherwise deserted streets of the capital. Many are angry at Mursi'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 Mursi 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 in over a week of street violence in which hundreds have been hurt and several killed, including an American student.

Over 10,000 Mursi supporters also congregated in the capital, by a mosque not far from the suburban presidential palace. Mursi himself is working elsewhere. But liberal protest organisers plan a sit-in outside the palace from Sunday evening.

Interviewed by a British newspaper, Mursi 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, which fueled 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. But the opposition dismissed it as too little to late. They hope Mursi 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 Mursi over the economy, many others fear that more turmoil will make life worse.

Mursi and the Brotherhood can hope protests fizzle out like previous outbursts. If they do not, some form of compromise, possibly arbitrated by the army, may be on the cards.

VIOLENCE

Both sides insist they plan no violence but accuse the other - and agents provocateurs from the old regime - of planning it.

Helicopter gunships flew over Cairo. The U.S.-equipped army, though showing little sign of wanting power, warns it may step in if deadlocked politicians let violence slip out of control.

U.S. 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.

In an interview with London's Guardian newspaper, Mursi repeated accusations against what he sees as attempts by entrenched interests from the Mubarak era to foil his attempt to govern. But he dismissed the demands that he give up and 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".

Liberal leaders say nearly half the voting population - 22 million people - has signed a petition calling for new elections, although there is no obvious challenger to Mursi.

The opposition, fractious and defeated in a series of ballots last year, hope that by putting millions on the streets they can force Mursi to relent and hand over to a technocrat administration that can organise new elections.

"We all feel we're walking on a dead-end road and that the country will collapse," said Mohamed ElBaradei, a former U.N. nuclear watchdog chief, Nobel Peace Prize laureate and now liberal party leader in his homeland.

ARMY ROLE

Religious authorities have warned of "civil war". The army insists it will respect the "will of the people".

Islamists interpret that to mean army support for election results. Opponents believe that the army may heed the popular will as expressed on the streets, as it did in early 2011 when the generals decided Mubarak's time was up.

A military source said the army was using its helicopters to monitor the numbers out on the streets. Its estimate on Tahrir in mid-afternoon was 40-50,000, with a few thousands at similar protest sites in other major cities.

It put the number at the Islamists' Cairo camp at 17,000. Having staged shows of force earlier this month, the Brotherhood has not called on its supporters to go out on Sunday.

Among the Islamists in Cairo, Ahmed Hosny, 37, said: "I came here to say, 'We are with you Mursi, with the legitimate order and against the thugs'.

"This is our revolution and no one will take it from us."

At Tahrir Square, banners ranged from "The Revolution Goes On", "Out, Out Like Mubarak" to "Obama Backs Terrorism" - a reference to liberal anger at perceived U.S. support for Mursi's legitimacy and its criticism of protests as bad for the economy.

"I am here to bring down Mursi and the Brotherhood," said Ahmed Ali al-Badri, a feed merchant in a white robe. "Just look at this country. It's gone backwards for 20 years. There's no diesel, gasoline, electricity. Life is just too expensive."

The Egyptian army, half a million strong and financed by Washington since it backed a peace treaty with Israel three decades ago, says it has deployed to protect key installations.

Among these is the Suez Canal. Cities along the waterway vital to global trade are bastions of anti-government sentiment. A bomb killed a protester in Port Said on Friday. A police general was gunned down in Sinai, close to the Israeli border.

Observers note similarities with protests in Turkey this month, where an Islamist prime minister with a strong electoral mandate has been confronted in the streets by angry secularists.

For many Egyptians, though, all the turmoil that has followed the Arab Spring has just made life harder. Standing by his lonely barrow at an eerily quiet downtown Cairo street market, 23-year-old Zeeka was afraid more violence was coming.

"We're not for one side or the other," he said. "What's happening now in Egypt is shameful. There is no work, thugs are everywhere ... I won't go out to any protest.

"It's nothing to do with me. I'm a tomato guy." (Reporting by Asma Alsharif, Alexander Dziadosz, Shaimaa Fayed, Maggie Fick, Alastair Macdonald, Shadia Nasralla, Tom Perry and Yasmine Saleh in Cairo, Yusri Mohamed in Ismailia and Abdelrahman Youssef in Alexandria; Writing by Alastair Macdonald; Editing by Mark Heinrich and Anna Willard)