"The government of Prime Minister Essam Sharaf has handed its resignation to the Supreme Council of the Armed Forces," Mr Hegazy said, in a statement carried by the official Mena news agency.
"Owing to the difficult circumstances the country is going through, the government will continue working until the resignation is accepted."
A military source told the BBC that the council was meeting to discuss the cabinet's offer, but there was still no consensus on whether to accept it. The same source said that the council was also consulting with other political groups.
The BBC's Yolande Knell in Cairo says the crowds in Tahrir Square cheered and shouted "God is great" when they heard of the resignation.
However they soon resumed an earlier chant of "the people want the removal of the marshal" - a reference to Field Marshal Mohamed Tantawi, who heads the Supreme Council of the Armed Forces.
He has the task of overseeing the country's transition to democracy after three decades of autocratic rule under Mr Mubarak.
The ruling generals are the real focus of demonstrators' anger, our correspondent adds. The interim cabinet is seen as having little power.
Burning barricadesAs night fell in Cairo, thousands more flocked to the symbolic square - the focal point of the protests which overthrew President Hosni Mubarak in February.ontinue reading the main story
Earlier, protesters had set up burning barricades in and around the square and thrown stones at riot police and troops.
Security forces responded with batons, tear gas and birdshot and fought running battles with the protesters.
Medics told the BBC they were seeing people injured by tear gas and rubber bullets.
Officials confirmed on Monday that more than 20 people had been killed and about 1,800 injured since Saturday.
Protests were also reported in other Egyptian cities, including in Alexandria where police are said to have fired tear gar to protect the offices of the security forces.
As the violence and tension escalated, the US called for restraint on all sides. "We're deeply concerned about the violence," said White House spokesman Jay Carney.
'Counter-revolution'The trouble started on Saturday following demonstrations against proposed constitutional changes unveiled by the interim government.
The military council produced a draft document setting out principles for a new constitution, under which the military and its budget could be exempted from civilian oversight.
A proposal by the military to delay the presidential election until late 2012 or early 2013 has further angered the opposition.
Many people fear the military plans to hold on to the reins of power, whatever the outcome.
Late on Monday, a coalition of political groups accused the military council of leading a "counter-revolution".
They called for a mass demonstration in Tahrir Square on Tuesday.
One of the protesters in Cairo, Ahmed Imam, 33, said handing power to the military after the overthrow of Mr Mubarak had been a mistake.
"We should not have left the streets. We handed power to the military on a silver platter," he said.
"The revolutionaries went home too soon. We collected the spoils and left before the battle was over."
Human rights watchdog Amnesty International has strongly criticised Egypt's military rulers, saying they had "completely failed to live up their promises to Egyptians to improve human rights".
