Reuters witnesses, a war monitor and state television reported intense fighting including artillery bombardment and small arms fire.
The army had made broad advances, said state television. The monitor, the Syrian Observatory for Human Rights, said it had gained control of several buildings in the densely built-up area.
Footage on state TV showed tanks rolling across an open area of fields to the edge of the enclave, which includes parts of al-Qadam district, al-Hajar al-Aswad and the Yarmouk Palestinian refugee camp.
It showed uniformed soldiers moving through battered streets with dense clouds of black smoke overhead, while the whizz and crash of artillery fire, the rattle of small arms fire and deep echoing blasts could all be heard.
Syrian President Bashar al-Assad this month defeated rebels in their biggest stronghold near Damascus in eastern Ghouta, and has since then focused on ending resistance in several smaller pockets near the Syrian capital.
Islamic State lost most of its territory in Syria last year in the face of two rapid offensives, one by the Syrian army backed by Russia and Iran, and the other by an alliance of Kurdish and Arab militias backed by the United States.
It has held onto some areas of desert in eastern Syria, as well as to the pocket in south Damascus and one other area by the borders with Jordan and Israel which is held by a group that has pledged allegiance to it.
Russia’s entry into the war in 2015 has propelled Assad to a succession of victories that have dashed any rebel hopes of ousting him militarily, but rebel groups still hold large swathes of northwest and southwest Syria.
Russian Foreign Minister Sergei Lavrov said on Saturday after meeting in Moscow with his counterparts from Turkey and Iran that the three countries need to help Syria’s government clear its country of terrorists.
Turkish Foreign minister Mevlut Cavusoglu meanwhile said Russia, Iran and Turkey needed to work with the United Nations to ensure the legitimacy of any political solution in Syria as any military solution would be illegal and unsustainable.