A spokeswoman for the Geneva-based ICRC, Krista Armstrong, told AFP that the 20,000 figure was an estimate and that the situation remained fluid, stressing that "people are fleeing in different directions", desperately seeking refuge from the brutal fighting.
Terrified civilians have fled empty-handed into remaining rebel-held territory, or crossed into government-controlled western Aleppo or Kurdish districts.
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The United Nations humanitarian chief Stephen O'Brien had earlier put the number of displaced from eastern Aleppo at 16,000.
The Syrian government offensive to recapture rebel-held parts of Aleppo has sparked international alarm.
France called for an immediate UN Security Council session on the fighting, which has seen the army capture a third of opposition-controlled east Aleppo in recent days.