The deal brokered by regime ally Russian Federation would see fighters with the Jaish al-Islam rebel faction leave Ghouta's main town of Douma for opposition-held territory in northern Syria, the Syrian Observatory for Human Rights said.
The eastern Qalamoun rebel enclave includes a few towns and an expanse of desert territory.
Syrian state media say the largest rebel group in Damascus' eastern Ghouta, the Army of Islam, has begun to evacuate from the area's last holdout town.
A human rights group says the fighting in the area has killed more than 1,600 people over the past one and a half months.
The pan-Arab al-Mayadeen TV said the agreement of Douma could be included in the next hours.
"This operation was part of the Coalition's mission to defeat ISIS, and we remain focused on our mission", Rankine-Galloway said.
The fighters in Douma were following their Faylaq al-Rahman comrades trapped in the other two Ghouta pockets that relocated to northern Syria last week.
The US has more than 2,000 military personnel in eastern Syria as part of worldwide efforts to defeat the Islamic State group, an extremist organization that once controlled swathes of Syria and neighboring Iraq.
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.
In video footage published by Jaish al-Islam online on Sunday, the group's leader told a group of men in a mosque he would stay.
In the spring of 2015, it held a massive military parade that included thousands of opposition fighters marching in formation and a striking display of tanks and armored vehicles at the doors of the Syrian capital.
Both moves follow Trump's decision in July to end a covert Central Intelligence Agency program under President Barack Obama to train moderate rebels to fight Syrian President Bashar Assad, marking an end of US efforts to remove the Syrian dictator.
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.
While the initial response of the military-intelligence agencies was to dismiss Trump's comment as though it was a random tweet, the remark was followed by reports revealing that the White House had put on hold some $200 million in State Department funding for "stabilization projects" in Syria.
It is also unclear whether all of the Army of Islam fighters have agreed to leave and how long the evacuations will take.
Council member Yad Abdelaziz said there was no such agreement in place, but that "humanitarian cases" would be allowed to evacuate on Monday.
Dozens if not hundreds of residents are believed to require care for war wounds and medical conditions exacerbated by the siege of the town.
The aid package being suspended was a priority of now-fired Secretary of State Rex Tillerson, who had emphasized the importance of a strong American presence in Syria as a way to stabilize the region and combat Islamic State-backed terrorists from regaining strength.