Free Syrian Army rebels are fighting along three fronts in the suburb of Jawbar, CNN reported.
"If we capture the area it means we have reached the heart of the capital," said Baraa, a spokesman for the Revolutionary Military Council who gave only his first name.
"It means we can cross into Damascus," he said.
This, as a foreign official said the Assad regime is ready to begin formal peace talks in the two-year-old conflict.
Iran's foreign minister said "there is no military solution" to the conflict.
"The Syrian government has voiced its preparedness (for talks) and there remains no excuse (for the opposition to avoid talks). Mr. Mualem has announced in Syria that his country's leaders are prepared to even talk with the armed opposition," Iranian Foreign Minister Ali Akbar Salehi said.
He told Iran's Fars News Agency the rebel forces would be "held accountable" if they continued to fight rather than negotiate with the Assad regime.
The FSA is now in control of a majority of Jawbar, the opposition Damascus Media Office said. The Assad regime is shelling the suburb with fighter jets and missiles.
Syrian state media said government forces had inflicted heavy losses in the Damascus suburbs and were pursuing armed terrorist groups in the area.
Fighting began Saturday morning in al-Raqa province near the Syrian border with Turkey, said the Syrian Observatory for Human Rights.
Some 125 people were killed across Syria on Friday, the Local Coordination Committees for Syria reported.