"For the sake of humanity, we call on, we plead with the parties and those with influence to do everything in their power to protect civilians and enable access to the besieged part of eastern Aleppo before it becomes one giant graveyard," UN humanitarian chief Stephen O'Brien told council members during the meeting.
French Foreign Minister Jean-Marc Ayrault called for the meeting a day earlier. He told Reuters that Syria's brutal civil war, which has raged for almost six years, would not be resolved with one of the "biggest massacres on a civilian population since World War Two."
"This (Security Council) meeting would have to find a way to deal with the humanitarian situation and see how we can get aid in. We have to find a way," Ayrault said.
Syrian regime forces entered eastern Aleppo by ground Saturday in an operation to wrest control of the area from rebels, who have ruled there for more than four years.
Civilians are fleeing by the thousands -- many to government-controlled western Aleppo -- as the regime pounds the east with airstrikes and rebels fire back from their strongholds.
Around 20,000 people have fled from eastern neighborhoods in recent days, O'Brien told CNN on Tuesday night, but around 200,000 remain.
The area has become a wasteland as a government siege has left it short of food, clean water, fuel and medical supplies. The United Nations is pushing for a humanitarian pause to bring in desperately needed aid. Bettina Luescher, a spokeswoman for the UN World Food Programme, said Tuesday that Aleppo civilians were enduring a "slow-motion descent into hell."
Around 600 people have been killed in eastern Aleppo since Saturday, activist sources have told CNN, including 45 on Wednesday, according to the Aleppo Media Center activist group. In western Aleppo, eight others were killed Wednesday, the state-run Syrian Arab News Agency reported.
Pressure on Russia
International calls are growing louder for a political solution to end the war -- something that has evaded the Security Council for years, with Russia using its veto power as one of five permanent members to block resolutions.
Russia is the most powerful ally of Syrian President Bashar al-Assad's regime and has carried out airstrikes since September 2015 to prop up the embattled leader. But Moscow has tried to distance itself from the current assault, saying Wednesday it hasn't bombed the city since October 18.
The Russian Defense Ministry said Wednesday that the Syrian army had cleared Castello Road, a key artery and supply line in Aleppo, so the United Nations can deliver humanitarian aid.
But UN humanitarian adviser Jan Egeland said the United Nations already is bringing aid to parts of western Aleppo, and that Castello Road "does not give access to the besieged parts of east Aleppo."
British Prime Minister Theresa May on Wednesday described the situation in Aleppo as "horrific" and called on Russia to act.
"We want to see a cessation of hostilities. We want to see opportunity for humanitarian aid to have access to Aleppo, and we will be pressing for that at the UN Security Council," May said in Parliament.
Egeland told CNN's Christiane Amanpour on Tuesday that the United Nations had met with Russian diplomats and the military to find a way to deliver aid.
"I believe we will get the pause that we ask for," he said, adding that officials were still waiting for a reply.
Despite criticism that the Security Council has failed the people of Aleppo, some residents said they remained hopeful an agreement will stop the violence.
"We are waiting to see if a deal would be struck to stop the shelling," Ahmad Edrees told CNN.
"People here are desperate. They want some hope."
What will Trump do?
The United States has been Russia's fiercest critic at UN meetings -- on opposing sides with its old Cold War enemy in the conflict. Moscow considers all opposition rebel groups as terrorists, while Washington has supported and armed some of the same groups.
Questions have swirled over what approach US President-elect Donald Trump will bring to the war. As a candidate, Trump praised Russian President Vladimir Putin and suggested he wanted to improve US-Russia ties.
The Kremlin did not confirm Russian media reports that Moscow had communicated with the incoming Trump administration on Syria. But Putin said in a speech Wednesday that he had spoken with Trump and they agreed their countries' "unsatisfactory" relationship had to be "straightened out."
Assad has vowed to take all of Aleppo, and regime forces have made rapid progress since entering the east
last weekend. They have seized control of a huge portion of the northeast.
Russian media have said that around half of the territory is now in regime hands; however, CNN sources put the figure at a little more than 20%. Their gains drive a wedge through the zone and split it into two sections.
Taking all of Aleppo -- the last urban rebel stronghold in Syria -- would mark a turning point in the war. Seizing it would put the Syrian regime back in control of all four major cities and make an opposition government less likely.