In a tweet, it said that “critical” patients were being moved from the Eastern Ghouta area to the capital.
Last week, a UK charity said President Bashar al-Assad had been considering a request to evacuate seven children with cancer from the area, which has been under government siege for four years.
They are among more than 130 children needing urgent medical treatment there.
Nearly 12% of children in Eastern Ghouta – which has a total population of 400,000 – are suffering from acute malnutrition, the UN has said.
On Tuesday evening, the International Committee of the Red Cross published pictures of ambulances brought in to carry out the evacuations.
It provided no details about how many people would be moved out of the area.
The Syrian Arab Red Crescent later posted images of several young children accompanied by their mothers, saying they would be taken to hospitals in Damascus.
Last week, Hamish de Bretton-Gordon, an adviser to the Union of Medical Care and Relief Organisations (UOSSM) charity, told the BBC that President Assad’s private office had said he would consider the evacuations of children with cancer.
“We understand Assad is thinking about it. And we’re calling him back on Tuesday morning to speak to him direct,” said Mr de Bretton-Gordon.
“And if he gives us the go-ahead then the plan is that we will get to Ghouta as quickly as we can, get the children.”
The charity operates in Eastern Ghouta.
The Syrian government has not publicly commented on the latest developments.
Earlier this month the Red Cross said life in Eastern Ghouta was becoming “impossible” and the situation there had reached a “critical point”.
The UN has been trying for weeks to arrange medical evacuations.
Dozens of civilians are reported to have died in recent government bombardments and food shortages have led to severe malnutrition.
The area has been designated a “de-escalation zone” by Russia and Iran, the government’s main allies, along with Turkey, which supports the opposition. (Courtesy BBC)