Negotiations have been held between the armed group, the Army of Islam, and Russian Federation, supporting the Assad administration.
The Army of Islam has refused to respond to media requests for comment.
Trump has made clear that "once ISIS and its remnants are destroyed that the United States would be looking toward having countries in the region playing a larger role in ensuring security and leaving it at that", one official said.
The deal over Douma would mark the end of a weeks-long push by Assad's forces to consolidate their control over eastern Ghouta, just outside the capital.
Douma's fall will represent the rebels' worst defeat since 2016, driving them from their last big stronghold near the capital.
Backed by Russia, Assad's forces have scor-ed a series of victories over rebel forces in recent years.
Fighters from Jaish al Islam are reported to have reached an agreement to allow them to leave Douma, the last remaining suburb under rebel control in the suburb on the outskirts of Damascus.
"The fear is of the unknown and not bombardment or hunger. what the coming days will bring we can not predict", said the resident, Bilal Bwaidani.
Mr Assad's forces have retaken 95 percent of Eastern Ghouta since launching a blistering air and ground assault on the besieged enclave on February 18, killing 1,600 civilians and displacing tens of thousands more.
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.
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Although no official statement was issued by the faction, sources told Asharq Al-Awsat that the deal "is not similar to the previous ones that involved Harasta, which was controlled by Harakat Ahrar al-Sham, or Jubar and Zamalka, which were controlled by Failaq al-Rahman". The town is one of the last around the capital to hold out against the government.
Tens of thousands of people have now evacuated once-bustling towns in the suburbs east of the capital, which had almost two million people before the start of the conflict and were major commercial and industrial hubs. Talks with Jaish al-Islam for the evacuation of Douma had dragged on for several days, with residents growing nervous.
"Of course I would rather leave". More than 11 million people - about half the country's prewar population - have been displaced from their homes.
Syria's army promised Saturday to finish off fighters in the final opposition holdout of devastated Eastern Ghouta outside Damascus after a penultimate pocket was declared "empty" of rebels.
On Friday, Russia's defence ministry said it had brokered a deal that would see rebels abandon Douma "shortly", but the fighters denied it immediately.
"But if they give us the choice of dying or leaving, it will be another matter".
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It says a local council for Douma will be formed with the approval of the central government.
Buses entered the enclave on Tuesday to prepare for a second day of evacuations, a military source and the Britain-based Syrian Observatory for Human Rights monitoring group said.