Another rebel spokesman, Hamza Birqadar of the Jaish al-Islam group, said in a tweet on Thursday that insurgents had regained some positions in a counter-attack.
"We were taken aback by the fighting that broke out despite guarantees from the parties involved in this conflict that humanitarians could enter Douma, in eastern Ghouta", ICRC's Regional Director for the Near and Middle East, Robert Mardini said.
An aid convoy crossed into the rebel-held suburbs of Damascus on Friday, delivering desperately needed aid despite heavy fighting that broke out near the convoy and renewed airstrikes by the Syrian government.
Before the war, eastern Ghouta provided a good portion of the Damascus' food.
Residents say they fear government retribution if they were to try and flee the area, or the possibility of being redrafted into the Syrian military and sent to fight on the frontline.
However, UN agencies said most medical supplies had been stripped from the convoy by Syrian government officials and that the food supplies brought in were insufficient.
For the first time since Monday, food supplies got into the town of Douma, the Red Cross said.
Doctors Without Borders said Friday that between February 18 and March 3 at least 1,005 people were killed and 4,829 wounded - or 71 killed and 344 wounded on average per day.
Observatory head Rami Abdel Rahman said talks were underway between government representatives and local figures to organise the evacuation of civilians or fighters from parts of the enclave.
Rebels in recent days tried to slow the advance with an attempted counter-offensive, but were struggling to hold off President Bashar al-Assad's forces.
On March 3, regime forces - with Russian air support - launched a major ground offensive in Eastern Ghouta.
Eastern Ghouta's 400,000 inhabitants have lived under government siege since 2013 and the enclave - more than half of which has over the past three weeks been retaken by regime forces - is home to a myriad of armed groups.
Air strikes hit the areas of Jisreen and Harasta after stopping briefly in the early morning, the Britain-based Syrian Observatory for Human Rights monitoring group said. The roads connecting the towns to each other were all covered by army fire, the Observatory said.
A negotiator from the town said a "civilian delegation" was "to negotiate with the regime towards a solution to end the fighting" in the area.
That has in effect cut the large towns of Harasta and Douma off from each other and the rest of the enclave, it added.
The U.N. High Commissioner for Refugees, Filippo Grandi, said the United Nations has "failed spectacularly" when it comes to Syria.