"Around 400 families, approximately 2,000 people, were able to leave the camp on Friday and Saturday via two secure roads to the Zahira district, which is under army control," said Anwar Abdul Hadi, a Palestine Liberation Organisation official.
"The evacuations are continuing on Sunday and some people are now in Yalda (a district under regime control) and waiting to be taken to shelters."
State news agency SANA also reported the evacuations, saying they were carried out in coordination between the army and Palestinian factions and that the government had set up shelters for those fleeing.
Abdul Hadi said most of those evacuated from the camp were being hosted in government shelters, with at least 25 wounded taken to hospital.
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The Syrian Observatory for Human Rights, a Britain-based monitoring group, confirmed that "hundreds" of people had been evacuated from the camp.
It said at least 26 people, including civilians as well as fighters from IS and Palestinian factions, had been killed in the camp since Wednesday.
The UN agency for Palestinian refugees UNRWA said it had today given "life-saving humanitarian support to 94 civilians, including 43 women and 20 children, who fled Yarmuk."
Since the jihadist advance began Wednesday, regime forces have pounded the camp with shells and barrel bombs, according to the Observatory.
Palestinian officials have urged humanitarian access to the camp and called for its residents to be protected.
"The Palestinians are paying the price for wars and violence that are not of their making," Palestinian president Mahmud Abbas said in a speech today.
He said a solution must be found "to protect the residents of Yarmuk who have done nothing to deserve this."