Meanwhile figures close to rebels said talks were underway with the aim of them leaving to other opposition areas or giving up arms, though rebels have publicly ruled out the kind of negotiated withdrawal that helped Assad recover Aleppo, Homs and other areas.
Troops have splintered Ghouta into three besieged zones in one of the bloodiest offensives of the seven-year war, with rebels facing their worst defeat since the battle of Aleppo in 2016.
The Syrian government has sought to drive a wedge between rival rebels who control different parts of eastern Ghouta. By working on separate secret talks with rebels in each region by applying varying degrees of military pressure, the government's "divide and rule" tactics were beginning to bear fruit, one opposition source said.
"The army advances in recent days have piled pressure on the factions and every party wants to get the best deal possible," the source said without elaborating.
In a sign of confidence, Assad on Sunday visited frontline troops and people recently displaced from eastern Ghouta, Syrian state media said and published footage and pictures of the meetings.
"Every tank driver who advances one metre forward has changed the political map of the world," state news agency SANA reported Assad as saying.
After a morning of calm, shelling and ground battles resumed across eastern Ghouta on Sunday afternoon, the Syrian Observatory for Human Rights war monitor said.
On Sunday, Failaq al-Rahman, the main rebel group in the southern eastern Ghouta pocket said it was negotiating with a United Nations delegation about a ceasefire, aid and the evacuation of urgent medical cases.
"We are engaged in arranging serious negotiations to guarantee the safety and protection of civilians," said Wael Alwan, Failaq al-Rahman's Istanbul-based spokesman.
"The most important points under negotiation are a ceasefire, ensuring aid for civilians and the exit of medical cases and injured people needing treatment outside Ghouta."
But a general evacuation of civilians and rebels was not on the table, he said.
A number of patients requiring urgent medical attention have been evacuated from the northern pocket and some aid has entered there in recent days. This has not yet happened in the southern pocket.
Figures close to the two main rebel groups - Failaq al-Rahman in the southern pocket and Jaish al-Islam in the northern enclave - have told Reuters discussions are underway for the transfer of Failaq al-Rahman and Jaish al-Islam fighters to opposition held areas in northern and southern Syria respectively.
Also on Sunday, state TV said the Syrian army had given a group of rebels in another pocket, the smaller Harasta area, an ultimatum to withdraw.
An opposition official from Ghouta privy to the negotiations told Reuters a delegation had left Harasta and was working out the terms of a deal that would follow previous surrender accords for rebel-held areas subjected to bombardment and siege.
On Sunday afternoon he said there was an agreement in principle and it would be enacted within 24-48 hours.
Local rebels were offered two choices - either to leave to opposition areas in northern Syria or settle their situation and become part of local pro-government militias that would maintain security.
Hundreds of fighters have already handed themselves in during the last forty eight hours and begun reconciliation deals with the Syrian government, another rebel source in the area said.
Other terms of the deal would ensure that they handed over their weapons, he added.
CONDITIONS INSIDE "DIRE"
The United Nations resident coordinator in Syria, Ali al-Za'tari told Reuters on Sunday he estimates around 25,000 people have left eastern Ghouta in the past week.
Civilians have been making their way out towards army positions on foot, hauling their belongings with them. They are then transferred to one of three reception centres.
The Observatory said about 50,000 had left the southern pocket in the past 72 hours and thousands left on Sunday. Russian news agencies said more than 73,000 people have left eastern Ghouta so far, 25,000 of which left on Sunday.
The U.N. office for humanitarian affairs (OCHA) said in a statement on Thursday conditions for those remaining in eastern Ghouta were "dire".
"There is limited food, with reports of the usual ration of bread for one day being consumed over the period of a week to 10 days; insufficient sanitation and hygiene support for those living in basements, and increased risk of communicable disease," the statement said.
The U.N. is hoping another aid convoy of 25-30 trucks will enter Douma in the northern pocket in the next couple of days, al-Za'tari said.
"But this is a drop in the bucket of what people in Douma require," he said.