A Middle East diplomat described battles in the city as a "major engagement", with fighting going back and forth. He said: "The opposition is hitting Damascus from a multiple of directions and the regime is trying to stop it."
Jets bombed Jobar, a neighbourhood adjacent to the main Abbasid Square, and the suburb of Daraya on the highway to Jordan to the south, sources said.
The two areas are part of interconnected Sunni Muslim districts in and around Damascus that have been at the forefront of the 22-month uprising against four decades of Assad family rule.
Rebels entered Jobar last week after breaching the regime's defence lines at the ring road and taking several army and pro-Assad militia positions in the district.
The road, a supply line for elite army units dug in around the city centre, separates the capital from the mostly rebel-held expanse of Sunni towns and suburbs known as eastern Ghouta.
To the south-west, near the main highway to Jordan, heavy bombardment was reported on the suburb of Daraya, where the army advanced in the last few days, breaking a two-month rebel hold.