"Last Chance Friday... stability or chaos," ran the headline of the flagship state-owned daily Al-Ahram.
The call for protest comes amid uncertainty over who will lead the next government, after privately-owned television channels reported that former premier Kamal al-Ganzuri had been given the job.
There was no immediate official confirmation but Al-Ahram said on its website that Ganzuri had agreed to lead a government of national salvation.
Ganzuri is an economist who served as prime minister under Mubarak from 1996 to 1999.
If confirmed, the move would mark the latest attempt by the Supreme Council of the Armed Forces (SCAF) to regain the initiative after days of deadly clashes between security forces and protesters since Saturday.
The violence, in which at least 41 protesters have been killed -- 36 of them in Cairo -- and more than 2,000 injured, led to the resignation of caretaker premier Essam Sharaf's cabinet on Tuesday.
The call for new protests in Tahrir Square came despite an apology by the military rulers on Thursday for the deaths of protesters at the hands of police.
The violence has cast a shadow over Monday's parliamentary elections, prompting the SCAF to pledge to maintain security during the first poll since a popular uprising toppled Mubarak in February.
There was an uneasy calm on Thursday among the crowds massed in the square, the epicentre of Egypt's popular unrest, after a truce negotiated by Muslim clerics.
The cabinet said in a statement posted on Facebook that an agreement had been reached between security forces and protesters to avoid further confrontations.
But the military council ruled out stepping down.
"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.
Another senior SCAF member, Major General Mamduh Shahine, also insisted they would not change course.
"We will not delay the elections. This is the final word. They will be conducted according to the original dates."
Mulla said the SCAF, the armed forces and the interior ministry would work together to guarantee the security of the elections.
There were fresh clashes between security forces and protesters in the Mediterranean city of Alexandria Thursday, the state MENA news agency reported. Two people have died there since Saturday. There was also unrest in the Nile Delta region.
Some activists have called for the protests to be focussed on the capital -- and some argue that demonstrators should stop provoking police guarding the interior ministry building in Alexandria.
In Tahrir Square, some demonstrators remained unequivocal in their demands for the military to go.
But many Egyptians are worried about the impact on the country of the standoff between the protesters and the army.
"If the military just steps down, there will be chaos. I mean, there is chaos now, so imagine what would happen if the military steps down," said Essam al-Arabi, who sells leather handbags near Tahrir Square.
Standard and Poor's said on Thursday that it had cut its long-term rating on Egypt by one notch to 'B+'.
"The downgrade reflects our opinion that Egypt's weak political and economic profile... has deteriorated further" following the latest clashes, the ratings agency said.
Tantawi, Mubarak's long-time defence minister now in charge of the country, tried to appease protesters and defuse Egypt's political crisis in a rare television speech on Tuesday.
But he was criticised for not mentioning the deaths at the hands of police.
The SCAF vowed on Thursday to investigate and prosecute all those behind the deaths, and to offer assistance to the families of the dead and injured.
The deaths had prompted an unusually strongly worded statement from Al-Azhar, Sunni Islam's highest seat of learning, denouncing police violence.
Any dialogue "stained with blood is doomed and its fruit will be bitter", grand imam Sheikh Ahmed al-Tayyeb said.
Al-Azhar "calls on the police leadership to immediately issue orders not to point their weapons at demonstrators... no matter what the reasons", Tayyeb said in a message aired on state television.
UN High Commissioner for Human Rights Navi Pillay called the images coming out of Egypt "deeply shocking" and urged the authorities to end their "clearly excessive use of force" against protesters.