Mr Al Assad and his allies say their offensive in Eastern Ghouta is necessary to end the rule of militants over the area's people, and to stop them shelling government areas.
The Central Military Media outlet, which is linked to the Syrian military, said that the rebel group had agreed to leave Douma for Jarablus, a town shared between rebel and Turkish control in northern Syria.
A UK-based monitoring group, the Syrian Observatory for Human Rights, said Sunday's deal would allow Jaish al-Islam fighters and their families, as well as civilians who wanted to leave, to travel to rebel-held areas in the northern province of Aleppo.
A map shows the advance of the Syrian military into eastern Ghouta outside of Damascus, Syria, as rebels and jihadis lose territory and entered into deals with the Syrian government. The town was one of the hubs of the Arab Spring uprising against President Bashar Assad's government in 2011, which drew a brutal response from security forces, sparking the ongoing civil war.
Douma is the last town to hold out against government forces in the once rebel-held suburbs.
Syria's army had on Saturday vowed to finish off rebels in Douma.
The last Syrian opppsition group in eastern Ghouta near Damascus began withdrawing on Monday under an agreement with the Assad regime, regime media said, though a military source said a group of opposition fighters were still rejecting the deal.
Jaish al-Islam has been in Russian-brokered negotiations with the Syrian government for several days now.
"Army units are continuing their combat operations to rid Douma of terrorists", the army statement said.
Successive evacuation deals were reached with rebel forces in parts of the enclave, with more than 45,000 fighters and civilians bussed out in recent weeks, according to figures provided by state news agency SANA.
Talks between the state and the Jaysh al-Islam insurgent group that holds Douma have so far failed, making further military action likely.
Late on Sunday, Russian news agency Interfax quoted General Yuri Yevtushenko as saying a "preliminary deal" had been reached to evacuate Jaish Al Islam fighters.
Defense analysts say a major goal of the government's campaign was to complete a security belt around the capital, where for years rebels dug into a network of tunnels and well-fortified positions resisted countless offensives to seize the enclave.
Backed by Russia, Mr Assad's forces have scored a series of victories over rebel forces in recent years, often through campaigns of siege, aerial bombardment as well as ground offensives that have drawn widespread global condemnation.
150,000 people displaced Tens of thousands of civilians remain in Douma, which is the largest urban centre in the enclave, facing worsening humanitarian conditions.