If confirmed, Mansour has tried to find a technocrat without the baggage of ElBaradei, whose candidacy outraged Salafi Islamists in a loose coalition that backed president Mohamed Morsi's overthrow by the military on Wednesday.
The Salafis say ElBaradei, viewed as an ardent secularist and top opponent to Islamist Morsi, would have been a divisive premier.
But the head of the Salafi Al-Nour party also objected to Bahaa Eldin, because the business lawyer used to belong to ElBaradei's National Salvation Front coalition.
"We don't object to (Eldin) personally, he is an economic pillar," Yunis Makhyun told the Dubai-based Al-Arabiya television channel.
"We reject his candidacy because he belonged to the National Salvation Front," he said.
Muslimani has said that while the presidency would listen to objections against prime minister candidates, it could not please everyone with its final choice.
The son of a prominent writer, Bahaa Eldin would be handed the enormous task of bringing a semblance of unity to the new Egypt, just days after the military ousted Morsi.
The development came as protests swelled to an estimated 250,000 in Cairo's Tahrir Square, epicentre of the 2011 revolution which toppled Hosni Mubarak.
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" Morsi 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 Morsi, 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 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-Morsi 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 coast, 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.
Morsi'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.
In an interview published on Sunday, ElBaradei called for "inclusion of the Brotherhood in the democratisation process".
"No one should be taken to court without a convincing reason. Former president Morsi must be treated with dignity," the former UN nuclear watchdog chief told German news weekly Der Spiegel.
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.
At least 37 people died in violence following Friday's Islamist rallies despite talk of peaceful protests, with Cairo and second city Alexandria the hardest hit.
The bloodletting continued on Saturday, with gunmen killing a Coptic priest in the Sinai and other militants 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.