State television and Syrian newspapers said information indicated that, under the deal, the group would hand over heavy and mid-sized weapons and acknowledge the restoration of state sovereignty over Douma.
The child, who was just five weeks old when he was injured, will "survive on behalf of all innocent children of Syria" and "remind us that we have a conscience as humanity", he added.
On Sunday, the Syrian government-linked Central Military Media outlet said that once the evacuations are completed, a local council for Douma will be formed with the approval of the central government. By midday, only a handful had left for the north in three buses.
There was no immediate comment on Monday from the rebels themselves.
A civilian committee taking part in the negotiations with Russian Federation said a deal had been reached "to evacuate humanitarian cases to northern Syria".
The Army of Islam, the largest rebel group in Syria, is fleeing Eastern Ghouta.
Syria's state news agency SANA said 24 buses carrying around 1,200 fighters from Failaq al Rahman and their families left Douma on Sunday in the first such pullout of fighters.
If confirmed, the departure of Jaish al-Islam from Douma would mark the end of the war for eastern Ghouta, wiping out an opposition stronghold near Damascus and underlining President Bashar al-Assad's unassailable position in the war.
Pope presides over Good Friday amid increased security
Those marching said they want to call attention to the many social evils that still plague communities, all over the world. Another passer-by, Eric, who was also happy to see the re-enactment, said: "Is come I come up and buck up inna this".
"Jaish al Islam has taken the decision to remain steadfast and the idea of leaving is not on the table", said the official.
A Syrian military source was quoted as saying that some of the fighters were rejecting the deal.
Syrian opposition groups have also fired mortars into Damascus. Backed by Russia, Assad's forces have scored a series of victories over rebel forces in recent years, often through campaigns of siege, aerial bombardment and ground offensives that have drawn widespread worldwide condemnation.
The Syrian army last week warned the insurgents to surrender or face a military assault to drive them out.
Douma was one of the Syrian cities that gave birth to the anti-Assad protests in 2011.
A resident reached by phone described a sense of foreboding among those still in the besieged city.
Despite the resolution, however, the regime and its allies early in March launched a major ground offensive - backed by Russian air power - aimed at capturing opposition-held parts of the district.
"If they intend to support terrorist elements or provide direct or indirect protection through their armed forces, this would be a truly calamitous step", Turkey's Defense Minister said during a visit to the country's northeastern province of Giresun on Sunday.