Rebels planned to push into the city center from their strongholds on the outskirts and fighting has been fierce. The army sent reinforcements after a week of rebel advances, including the capture of two military bases near the capital.
Activists said the heavy rocket attacks on Sunday killed at least ten in the town of Deir al-Asafir, 12 km east of Damascus. Video published by activists from the town showed at least five bodies, one of them a young boy and one an elderly man. The other bodies were wrapped in blood spattered white sheets.
Another video showed smoke rising over the skyline and some buildings in flames. Opposition reports are difficult to verify because the government restricts media access in Syria.
Syria's 20-month-old uprising has grown increasingly bloody in recent months, and activists say more than 40,000 have died. More than 200 people were killed on Saturday according to the opposition-linked Syrian Observatory for Human Rights, with at least 43 dead around Damascus and its suburbs.
Syrian security sources and diplomatic contacts told Reuters that Assad's forces launched an offensive this week in a move to seal central Damascus off from the suburbs.
The army's assault appears to have staved off a rebel advance into central Damascus so far. But neither side has gained ground in recent days, and fighting continues along the outskirts of the city despite heavy shelling by Assad's forces.
Clashes around Damascus International Airport went into their fourth day on Sunday. It has effectively been closed since Thursday when the army tried to push back rebels there who say the entire airport road had become a battleground.
"The Free Syrian Army is striking the reinforcements trying to enter the airport to help the regime's forces...there are clashes all along the airport road," said Abu Nidal, a rebel spokesman in Damascus, speaking by Skype.
Rebels say they want to control the airport because the army has used it to bring in weapons. Western intelligence reports earlier this year said that Iran, Assad's main backer, had been using civilian aircraft to fly military equipment and personnel through Iraqi airspace into Syria.
American officials say that arms flow has continued due to Iraqi reluctance to check flights, according to a New York Times article. It said only two inspections had occurred since Iraq agreed to a US request in September and that Iran may have been tipped off about the searches.
Prime Minister Nuri al-Maliki told reporters in a press conference in Baghdad that there was no such request.
"There is no ability to inspect all planes destined to Syria and there was no US request to inspect all aircrafts because they know that this is not possible," he said on Sunday.
"We said to the Syrians, Iranians, U.S, UN and the world, we said as the Iraqi Government, we are committed to (prevent) the weapon ... We prevent the passage of arms for you because our constitution states that Iraq is not a pathway or a seat for any similar actions".Car bombs
In Syria's central city of Homs, a car bomb killed at least 15 people and wounded 24 on Sunday, Syria's state news agency SANA said. It said the blast in the city's Hamra district also damaged many nearby residential buildings.
There has been a rise in the number of car bombs around the country. The British-based Observatory, which has a network of activists across Syria, reported four car bombs on Saturday.
Violence has risen in Syria particularly since rebels began to contest Assad's control around the capital and Syria's largest city Aleppo, but foreign powers remain deadlocked.
Western countries support the opposition but Russia, Syria's main arms supplier, and China have blocked three UN Security Council resolutions condemning Assad and reject sanctions on his government.
Assad, whose family has ruled Syria for four decades, says he is fighting off radical Islamist militants funded by the West and Gulf Arab countries.
State television on Sunday said the army had been "eliminating al-Qaida terrorists" in several suburbs surrounding Damascus including the rebel stronghold of Daraya.
The army entered part of Daraya, rebels said, a suburb on the southern outskirts of Damascus where fighters have launched mortars into the city. Rebel spokesman Abu Nidal said the army had entered one side of the town but that rebels were still in control of the rest of the area and were fighting back.
The army was firing heavy artillery and rockets into the town, rebels said.
BEIRUT (Reuters) -- Syrian army forces pounded rebel-held suburbs around Damascus with fighter jets and rockets on Sunday, activists said, killing at least ten and wounding dozens in an offensive to stop rebels closing in on the capital.