Earlier on Monday, Turkey's army asserted that it had taken control over half of the area under rebel control.
After pro-Syrian government militia joined the battle in Afrin on behalf of the Kurds, Ankara slammed the move, saying that the fighters had become Turkey's "legitimate target".
Turkey launched its operation "Olive Branch" in northern Syria almost two months ago to sweep the Syrian Kurdish YPG from the Turkish border.Turkey sees the YPG as a terrorist group and an extension of the outlawed Kurdistan Workers Party (PKK).
But the YPG has been a key ally of the United States in the fight against jihadists in Syria and Turkey's operation against it has raised tensions with Ankara's North Atlantic Treaty Organisation allies in Washington and Europe.
Sealing off Afrin city would be major for Turkey's "Olive Branch" operation, launched on January 20 and aimed at ousting the YPG, a vital partner for a US-backed coalition against the Islamic State group, but seen by Ankara as "terrorists".
A joint Turkish and Syrian rebel attack on the Kurdish enclave of Afrin has drawn Kurdish fighters away from the USA -led coalition fight against ISIS and into an alliance with Assad against Turkey.
The air strike also wounded two Kurdish fighters.
The Turkish military announced Monday evening that they have fully surrounded the Syrian city of Afrin.
It's a major issue of conflict for Turkey and the US that is expected to be discussed between the two countries later this month.
Russian news agencies say at least 100 civilians have been evacuated from a rebel-held enclave outside the Syrian capital.
The area has been under government siege since 2013, leaving hospitals too strapped to treat people properly.
Erdogan's foreign minister, Mevlut Cavusoglu, said last week that Turkey and Iraq's central government could start a joint military operation against Kurdish militants after Iraqi parliamentary elections scheduled for May 2018. Damascus has dubbed the Turkish campaign as "aggression", however, refrained from sending armed forces to the area.
Hamza Bayraqdar, the group's chief military spokesman, said in a video statement posted online that "our revolutionary ideology does not allow us to sell the blood of the holy warriors who liberated Ghouta".
More than 1,180 civilians have been killed since the Ghouta assault was launched.