President Donald Trump and other leaders requested that the rescue workers be evacuated.
Two different offensives in recent days have prompted an exodus of thousands of civilians.
The onslaught has been one of the fiercest of the civil war, now entering its eighth year.
The U.N. Security Council approved a resolution demanding a 30-day truce to allow aid deliveries and medical evacuations.
More than 300 have been killed since Sunday.
Only four people have been transported to Damascus hospitals so far. Others have died waiting.
At least 350,000 trapped Syrians lack access to food and medicine.