The Syrian Observatory for Human Rights said regime forces had advanced to 3km from Douma, the main town, after retaking "more than 25 per cent" of Eastern Ghouta, in operations mostly through farmlands.
Almost 600 people have been reported killed and more than 2,000 injured in air and ground-based strikes since 18 February, UN regional humanitarian coordinator Panos Moumtzis said, noting that mortar shells fired from the rebel-held enclave into the capital Damascus had killed and injured scores of civilians.
Speaking to a small group of reporters in Damascus, Assad said the five-hour daily "humanitarian pause" in eastern Ghouta would continue, to allow for any civilians wishing to leave the area to do so. But while the air campaign has eased, fighting has intensified on the ground.
Fighting rocked Syria's Eastern Ghouta on Sunday as rebels battled to retake territory seized in a devastating regime assault on the last major opposition enclave near Damascus.
Regime forces gained control of villages on the eastern side of the besieged enclave, which it said were controlled by Al-Nusra Front, a former Al-Qaeda affiliate.
They have reached the centre of the enclave, to the edge of Beit Sawa, according to the Observatory.
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After advances in recent days that saw the regime seize control of 10 per cent of Eastern Ghouta, rebel fighters clashed with regime forces on Sunday in the eastern part of the enclave, the Observatory said.
While the White House accused the Syrian government of a "continued use of chemical weapons", the Syrian and Russian militaries have repeatedly warned that the militants of Jabhat Al-Nusra, Faylaq al-Rahman and Ahrar al-Sham are plotting provocations using toxic agents so the West could pin the blame on the Syrian government. Damascus and Moscow say they are trying to clear the area of "terrorists".
The spokesman for the rebel Jaish al-Islam group, writing on Twitter, said government forces had mounted a failed attempt to storm a nearby area, al-Rayhan, but did not mention al-Shayfouniya. The U.N. said it would deliver humanitarian aid to the Eastern Ghouta enclave on Monday.
The Observatory said some 2,000 civilians had fled regime shelling and clashes in eastern areas to western parts of the enclave.
"Everyone is on the road".
"Many families are trapped under rubble, the rescue workers just can't cope".
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The intense bombardment of Eastern Ghouta has caused thousands of residents to flee their homes and head westward where the fighting is less severe, civilians inside the suburb told CNN Sunday.
"MORE FIGHTING, MORE DEATH " With no sign of meaningful Western pressure to halt the offensive, eastern Ghouta appears on course to meet the same fate as other rebel areas retaken by Assad, such as eastern Aleppo, which he recovered using similar tactics in 2016.
At the same time, ground-based strikes and mortar shelling from Eastern Ghouta have killed and injured scores of civilians in neighboring Damascus.
The violence comes on top of critical shortages of food and medicines and a catastrophic collapse of medical care amid the targeting of hospitals and first responders.
Earlier, a United Nations official in Syria told Reuters a humanitarian convoy carrying life-saving supplies from United Nations and other aid agencies would not enter Eastern Ghouta as had been planned on Sunday, citing a lack of permission.
"The convoy to east Ghouta is not able to proceed today", the official said, adding that the United Nations and its humanitarian partners "remain on standby to deliver desperately needed assistance as soon as conditions allow".
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