Tens of thousands of demonstrators rallied in Cairo's Tahrir Square on Friday demanding the end of military rule, following a week of deadly clashes, as Washington called for a swift handover of power.

Ahead of elections due to start on Monday despite the political turmoil, Egypt's ruling military council tasked Kamal al-Ganzuri, 79, a premier under ousted president Hosni Mubarak, to head a new cabinet.

With both sides trying to seize the initiative in the tussle between demonstrators and the military, the grand imam of Al-Azhar, Sunni Islam's highest authority, threw his weight behind the protesters.

The rally came just three days before the first parliamentary elections since Mubarak was toppled by a popular uprising in February and the military council took power.

"We believe that the full transfer of power to a civilian government must take place in a just and inclusive manner that responds to the legitimate aspirations of the Egyptian people, as soon as possible," White House press secretary Jay Carney said in a statement.

"The United States strongly believes that the new Egyptian government must be empowered with real authority immediately," Carney said.

The remarks came shortly before state television confirmed that the head of the Supreme Council of the Armed Forces (SCAF) had tasked Ganzuri, 79, with forming a new cabinet.

"Field Marshal Hussein Tantawi has decided to appoint Dr Kamal al-Ganzuri the new prime minister, granting him full powers," state television said.

In an uprecedented move by a figure who is appointed by Egypt's presidents, Al-Azhar's head on Friday expressed support for the protesters and wished them success, a senior aide said.

"The grand imam (Sheikh Ahmed al-Tayyeb) backs you and is praying for your victory," Hassan Shafie told the protesters during a visit to the square.

And the imam at the square who led tens of thousands of worshippers in prayer called on the ruling military to hand over power to a national salvation government.

Sheikh Mazhar Shahin said protesters would remain in the square until their demands were met.

"There is no option but a national salvation government with the powers of a president," said the charismatic Shahin, nicknamed the 'preacher of the revolution' after leading prayers throughout the revolt against Mubarak.

Former UN nuclear watchdog chief and presidential hopeful Mohamed ElBaradei, whose name has been widely touted to be part of a salvation government, joined the protesters in the square.

On Tuesday, the SCAF accepted the resignation of the caretaker cabinet headed by the once-popular Essam Sharaf, whose fall from grace was due to his perceived weakness in the face of the army.

The SCAF has said repeatedly that it does not have political ambitions and plans to hand power to an elected civilian authority after presidential elections which are set to take place no later than the end of June 2012.

On Thursday, SCAF insisted it would not bow to pressure from the protesters in Tahrir, saying they did not represent the whole country.

"The people have entrusted us with a mission and, if we abandon it now, it would be a betrayal of the people," senior SCAF member General Mukthar al-Mulla told reporters.

"The armed forces do not want to stay in power. We want to put the wishes of the people above all else," he said.

In Tahrir, among the tents of a week-old sit-in, protesters held what they called 'Last Chance Friday' while thousands attended a counter-rally in the Abassiya area, three kilometres (two miles) east of the square.

The latest Tahrir rally came as anti-military demonstrations enter their second week, despite an apology by the military rulers on Thursday for the deaths of protesters at the hands of police.

The violence, in which at least 41 protesters have been killed - 36 of them in Cairo - and more than 3,000 injured, led to the resignation of caretaker Sharaf's cabinet.

The deaths had prompted an unusually strongly worded statement from Al-Azhar, which denounced police violence and called on the interior ministry 'not to point their weapons at demonstrators'.