France has proposed that officials should monitor evacuation efforts and report on the protection of civilians.
It is expected to encounter resistance from Russia, which is an ally of Syria`s and a veto-wielding Security Council member.
Moscow has vetoed six resolutions on Syria since the conflict began in 2011.
France circulated a draft text late on Friday stating that the council is "alarmed" by the worsening humanitarian crisis in Aleppo, where "tens of thousands" are in peril.
Cold and hungry civilians remain stranded in the rebel-held east of the city, waiting to be moved to safety.
At least 6,000 people left east Aleppo under a fragile truce on Thursday, but the operation was halted a day later.
The besieged city has seen rapid government advances in recent weeks.
Aleppo evacuation : UN Security Council to vote on sending observers
18 December 2016
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The UN Security Council is to vote on whether to send observers to the stricken Syrian city of Aleppo, diplomats say.
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