Hukmiya Mohamed, a co-director of Roj camp, told Reuters that the 34 Australians had been handed to members of their families who had come to Syria for the release.
They were put on small buses for Damascus ahead of their departure from the country with a military escort.
However, the Australian families returned to Roj camp shortly after leaving, the sources said, due to "technical problems" between the families and the Damascus government.
The sources said that the families will head back to Damascus later on Monday.
"It's purely a procedural issue to be resolved today," a Syrian official said.
Roj camp holds more than 2000 people from 40 different nationalities, the majority of them women and children.
Thousands of people believed to be linked to Islamic State militants have been held at Roj and a second camp, al-Hol, since the jihadist group was driven from its final territorial foothold in Syria in 2019.
The Australian government said in a statement that it will not repatriate people from Syria.
"Our security agencies have been monitoring - and continue to monitor - the situation in Syria to ensure they are prepared for any Australians seeking to return to Australia," the statement said.
"People in this cohort need to know that if they have committed a crime and if they return to Australia they will be met with the full force of the law.
"The safety of Australians and the protection of Australia's national interests remain the overriding priority."
Syrian government forces seized swathes of northern Syria from the Kurdish-led Syrian Democratic Forces in January, before agreeing to a ceasefire on January 29.
The US military last week completed a mission to transfer 5700 adult male Islamic State detainees from Syria to Iraq.
Mohamed said the Syrian Kurdish-led authorities had previously facilitated numerous releases from Roj, via coordination with foreign governments.
Roj camp is where British-born Shamima Begum is held. The British government took away her citizenship on national security grounds in 2019.
"Shamima Begum's situation is the same as that of all the women in Roj camp ... If her country wants her back, our doors are open," Mohamed said.
