The forces under Syrian President Bashar Assad began their advance late Saturday in the region, the last major rebel stronghold near the capital, Demascus, Syria's Central Military Media said.
They have reached the centre of the enclave, to the edge of Beit Sawa, according to the Observatory.
Rami Abdel Rahman, the head of the Britain-based Observatory, said at least 12 government fighters had been killed in two areas, Al-Rihan and Shifoniya, in overnight clashes with the Jaish al-Islam rebel group.
At the same time, ground-based strikes and mortar shelling from Eastern Ghouta have killed and injured scores of civilians in neighboring Damascus.
Eastern Ghouta has been under siege for the last five years; humanitarian access to the area, which is home to 400,000 people, has been completely cut off.
Rebels have been forced to retreat and regroup in eastern Ghouta because of the Syrian government's "scorched earth" policy, a spokesman for the Jaish al-Islam rebel group said on Sunday.
The main rebel units actively holding territory in Eastern Ghouta are the Islamist Jaish al-Islam and Faylaq al Rahman, which have taken part in peace negotiations in the past. A reporter from the Syrian state-run Al Ekhbariya TV, accompanying troops, broadcast from Nashabiyah, a village on the south-eastern edge of eastern Ghouta.
Under siege since 2013, they had already been facing severe shortages of food and medicine.
"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.
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The Russian Defense Ministry confirmed the report, saying terrorists also opened sniper fire at the safe corridor, marked out by Moscow in Eastern Ghouta for evacuation of civilians. It accuses Moscow, which has been striving for a ceasefire and exit corridors for civilians, of "ignoring" a United Nations resolution.
The rebels say the humanitarian corridor is part of government efforts to forcibly displace the population, and have called on government forces to implement a full cease-fire adopted by the U.N. Security Council.
Macron's office said in a statement that he and Guterres spoke Saturday and expressed their "grave concern" and called for a full implementation of the resolution.
The UN's regional humanitarian coordinator for Syria, Panos Moumtzis, sounded the alarm on Sunday over the increase in violence.
Macron made a plea to Rouhani about the situation in eastern Ghouta because of the "links" between Iran and the Syrian government, the French statement said.
"Instead of a much needed reprieve, we continue to see more fighting, more death, and more disturbing reports of hunger and hospitals being bombed", Moumtzis said in a statement Sunday.
In a phone call, Macron and Rouhani agreed to work together in the coming days to obtain results on the ground, deliver the necessary assistance to civilians and implement the United Nations -sponsored cease-fire, the French president's office said in a statement.
Eastern Ghouta remains one of the few areas outside their control, along with the northwestern province of Idlib which is partly controlled by al-Qaeda-linked militants.