Egyptian president Hosni Mubarak said he fears chaos could break out in Egypt if he steps down.
As protests continued in Cairo tonight, Mr Mubarak said he would like to leave office but that he could not before elections were held in September.
"I am fed up. After 62 years in public service, I have had enough. I want to go," Mr Mubarak said in an interview with ABC news.
He said he was "unhappy" about the violent protests that broke out in the country for 10 days ago and he warned that the Muslim Brotherhood would fill any power vacuum left by his departure.
Gunmen fired on anti-Mubarak protesters in Cairo today and fighting has now killed at least six and wounded more than 800 there.
The continued unrest prompted new calls today from Western powers for Mr Mubarak to start handing over power immediately.
Thousands of angry young demonstrators on Tahrir Square said those who opened fire on them overnight were secret policemen. They insisted their resolve had only hardened. Rejecting Mr Mubarak's pledge on Tuesday to step down in September, they vowed to stand firm until the 82-year-old leader is gone.
Egypt's new prime minister Ahmed Shafiq apologised today for the violence. "As officials and a state which must protect its sons, I thought it was necessary for me to apologise and to say that this matter will not be repeated," the prime minister told reporters. Mr Shafiq said he did know who was behind the attacks and would investigate.
Opposition leaders including the liberal figurehead Mohamed ElBaradei and the mass Islamist movement the Muslim Brotherhood rejected a call to talks from Mr Mubarak's new prime minister, Ahmed Shafiq. Only the president's departure and an end to violence would bring them to negotiations, they said.
The protesters want free elections to tackle corruption and political repression which they blame for growing economic hardships similar to those frustrating people across the Arab world.
"One way or another we will bring Mubarak down," protesters chanted in the morning, amid rubble and burned out cars. "We will not give up, we will not sell out," others shouted.
Egypt's army and the Western powers which have supported it and Mr Mubarak as bulwarks against radical Islam have key roles to play, as the president's supporters seek to rally those Egyptians with much to lose from a collapse of the old order.
Many analysts see the army seeking to preserve its own 60-year-old position at the heart of secular Egyptian society by engineering a smooth removal of Mr Mubarak, a former air force commander. On Monday, it called protesters' demands legitimate and pledged not to open fire, given heart to the opposition.
But yesterday, troops stood by as Mubarak loyalists charged Tahrir Square on horseback and camels, lashing out at civilians. After dark, several demonstrators were shot dead. Not until this morning did soldiers set up a clear buffer zone around the square to separate the factions. Soldiers with a tank pushed back Mubarak supporters.
But that did not prevent new clashes, as groups pelted each other with rocks.
The United States, which gives Egypt's army some $1.3 billion a year, has made clear it wants Mubarak out, but has shied away from fully backing the opposition demands for him to leave the presidency immediately. The violence, however, has given fuel to international impatience with their Arab ally.
"This process of transition must start now," the leaders of Britain, France, Germany, Italy and Spain said in a statement.
They echoed the message US president Barack Obama said he gave Mr Mubarak in a phone call on Tuesday.
UN secretary-general Ban Ki-moon also said the transition of power in Egypt should begin now. "Many people are now asking the government should take reform. There have been calls for transition, very orderly peaceful transition. If a transition needs to be taken, the sooner the better. Transition should begin now,” he told a news conference in London.
The European Union's foreign affairs chief Catherine Ashton said her team had been calling Egypt overnight urging the army to protect the demonstrators and to give access for ambulances.
A senior US official said "somebody loyal to Mubarak has unleashed these guys to try to intimidate the protesters".
An Egyptian government spokesman called it a "fiction" to accuse ministers of orchestrating the violence. Demonstrators said over 100 attackers they seized were carrying documents associating them with the police or with Mubarak's ruling party.
"What happened yesterday made us more determined to remove President Mubarak," said a statement from protest movement Kefaya (Enough). "There will be no negotiations with any member of Mubarak's regime after what happened yesterday and what is still happening in Tahrir Square."
In a statement on Al Jazeera, the Brotherhood said: "We demand that this regime is overthrown and we demand the formation of a national unity government for all the factions."
Support for a new order is far from unanimous, however. The protesters in Tahrir Square, reduced largely to a youthful hard core including secular middle class graduates and mostly poorer activists from the Muslim Brotherhood, have been inspired by the example of Tunisia, where veteran strongman Zine al-Abidine Ben Ali was forced to flee last month.
But many other Egyptians have more respect for Mr Mubarak and seem willing to let him depart more gracefully in due course. Many fear anarchy. The secular-minded share concerns in the West, and in Israel, about the Muslim Brotherhood taking over.
An estimated 150 people have been killed so far and there have been protests across the country. United Nations human rights chief Navi Pillay said up to 300 people may have died.