CAIRO (AFP) - Opponents of Egypt's first freely elected leader packed streets across the country in their hundreds of thousands on Sunday to show the world his ouster was not a military coup but the reflection of the people's will.

Staged two days after Islamist rallies exploded into bloodshed, the protests came as a coalition that backed the military action to overthrow president Mohamed Mursi reportedly agreed to name a technocrat as premier.

Private television channel ONTV cited a presidential adviser as saying business lawyer Ziad Bahaa Eldin, 48, was set to be offered the post of interim prime minister.

If confirmed, caretaker president Adly Mansour has gone for a technocrat without the baggage of his original choice for the job, liberal leader Mohamed ElBaradei, who was now likely to become interim vice-president.

The development came as protests swelled to an estimated 250,000 in Cairo's Tahrir Square, epicentre of the 2011 revolution.

Wave after wave of military aircraft skimmed over the capital, with one formation leaving behind long trails of smoke in black, white and red - the colours of the Egyptian flag.

"We are on the street to show the world that it was a popular revolution and not a coup that overthrew" Mr Mursi on Wednesday, said a beaming teacher who gave her name as Magda.

Many banners showed the protesters' anger with the United States for what they perceive as its support for Mr Mursi, as well as American media coverage depicting his ouster as a coup.

"America shame on you! This is a revolution, not a coup!" read one, echoing a chant heard in Tahrir, again and again. Others carried portraits of army chief Abdel Fattah al-Sisi, the general behind Morsi's ouster.

President Barack Obama insisted overnight that the United States was "not aligned" with any political party or group in Egypt following Morsi's ouster.

"The future path of Egypt can only be determined by the Egyptian people," the White House quoted him as saying.

The Tamarod movement, which engineered the June 30 rallies that culminated in Morsi's overthrow, had led calls for people to gather at Tahrir and Ittihadiya presidential palace to "complete the revolution".

The anti-Mursi crowd filled the iconic square as people poured in from sidestreets, some unfurling a giant national flag emblazoned with the words "Go away" - a slogan used widely on June 30.

Bursts of gunfire were reportedly heard as people took to the streets of Alexandria on the Mediterranean and other major cities across the Arab world's most populous country.

Their Islamist rivals staged their own huge demonstrations in Cairo, where police armed with assault rifles watched over the pro-Morsi demonstrators.

Carrying pictures of the deposed president, the Islamists erected barricades across the capital, where tens of thousands of them blocked the main road to the international airport.

Mr Mursi's single year of turbulent rule was marked by accusations he failed the 2011 revolution that ousted autocratic president Hosni Mubarak by concentrating power in Islamist hands and letting the economy nosedive.

The rallies came after the coalition that backed Mr Mursi's ouster had wavered over the choice of Nobel Prize laureate ElBaradei as interim prime minister to lead the country out of the bloody crisis.

In an interview published on Sunday, Mr ElBaradei called for "inclusion of the Brotherhood in the democratisation process".

"No one should be taken to court without a convincing reason. Former president Mursi must be treated with dignity," the former UN nuclear watchdog chief told German news weekly Der Spiegel.

The official MENA news agency said on Saturday that interim president Mansour had appointed Mr ElBaradei, only for his office to later deny any final decision had been taken.

Salafi Islamists, who backed Mr Mursi's overthrow, were holding out against Mr ElBaradei's appointment, officials close to the talks told AFP.

But the 71-year-old was later tapped to become vice president, according to ONTV.

Morsi, who has been in custody since overnight on Wednesday, had issued a defiant call for his supporters to defend his "legitimacy" as Egypt's first freely elected president, in a recorded speech released shortly after his ouster.

Violence that killed at least 37 people following Friday's Islamist rallies came despite talk of peaceful protests, with Cairo and second city Alexandria the hardest hit.

The bloodletting continued, with gunmen on Saturday killing a Coptic priest in the Sinai and other militants on Sunday shooting dead a police officer in the restive peninsula.

Russia's President Vladimir Putin warned the stand-off threatened to degenerate into a civil war.

"Syria is already in the grips of a civil war, unfortunately enough, and Egypt is moving in that direction," news agencies quoted him as saying.