They include more than 300 militants from the extremist Ajnad al-Sham group and their families.
The European Union on Wednesday said it would chip in another three billion euros ($3.7 billion) for Syrian refugees in Turkey, and restrict travel for countries refusing to take back their citizens who fail to obtain asylum in Europe.
Washington's support for the Kurdish forces in that battle has infuriated Turkey.More news: KUDLOWMENTUM: Trump Drops Major Hint About Cohn Replacement
Turkey's repeated threats to push to Manbij have caused complications with North Atlantic Treaty Organisation ally the U.S., which has its troops deployed in the area and is backing the YPG in the fight against the Islamic State of Iraq and the Levant (ISIL, also known as ISIS), a move that has infuriated Ankara.
Since launching the offensive on the Afrin enclave on January 20, Turkey-led rebels have advanced to the outskirts of the city, sparking fears that it could become besieged.
According to The Independent's Patrick Cockburn, writing from Qamishli in northwest Syria, Turkey's surrogate FSA militia recruited Islamic State and al-Qaeda fighters who were expelled from Raqqa and eastern Aleppo.
US ambassador to the United Nations Nikki Haley has railed against the Syrian government's recently intensified campaign to retake the insurgent-held district of eastern Ghouta outside Damascus, accusing Syria and its Russian and Iranian allies of mounting civilian casualties.More news: Miley Cyrus sued for $300 million for allegedly stealing 'We Can't Stop'
The area has been under government siege since 2013, leaving hospitals too strapped to treat people properly.
Turkish president Recep Tayyip Erdogan on Wednesday said he hoped the northern Syrian town of Afrin would "fall by evening" to Turkish forces and rebel allies. It said the military had taken control of "critical areas" of the town but did not provide details.
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.
On Jan. 20, Turkey launched Operation Olive Branch to clear YPG/PKK-Daesh terrorists from Afrin.More news: Google Is Celebrating Pi Day With This Famous Chef's Pie Recipe
The largest rebel group in the besieged suburbs of the Syrian capital Damascus has vowed not to leave the area and to continue fighting advancing government forces.