Abbas asked U.N. Secretary-General Ban Ki-moon to intervene in the matter, the official Palestinian news agency Wafa said Wednesday.
"Mahmoud Abbas, the president of the state of Palestine, requested Wednesday that U.N. Secretary-General Ban Ki-moon and the international community enable our people in Syria to enter Palestinian territory," Wafa said.
The agency estimates some 450,000 Palestinian refugees live in 10 camps in Syria, some of which have been attacked in recent days.
Lebanese and Palestinian media reports estimate some 1,000 Palestinians fled to Lebanon following the attack on the Yarmouk refugee camp in Damascus earlier this week where dozens were killed and injured.
Following the attack, Abbas urged to keep the Palestinians out of the conflict in Syria where some 40,000 people have been killed since the violence between regime and rebel forces began in March 2011.
Wafa noted the Palestinian refugee camps in Lebanon are under the jurisdiction of the United Nations and are administered by the U.N. Relief and Works Agency for Palestine Refugees.