The reasons for the delay were not immediately clear. It came as shells fired by the Islamic State group on government-held parts of the eastern city of Deir el-Zour wounded two members of a Russian media delegation visiting the area, according to state-run Syrian news agency SANA.
Russia is a main backer of Syrian President Bashar Assad and Russian journalists enjoy wide access in government-held parts of the country.
The United Nations is not overseeing the transfer deal, which involves residents of the pro-government villages of Foua and Kfarya and the opposition-held towns of Madaya and Zabadani. All four have been under siege for years, their fate linked through a series of reciprocal agreements that the UN says have hindered aid deliveries.
Rami Abdurrahman, who heads the Britain-based Syrian Observatory for Human Rights, and Hezbollah's Al-Manar TV, earlier said that 3,000 people will be evacuated from Foua and Kfarya, while 200, the vast majority of them fighters, will be evacuated from Zabadani and Madaya.
It was not immediately clear if the evacuees feared attacks similar to Saturday's bombing.
Abdurrahman said Saturday's blast which hit an area where thousands of pro-government evacuees had been waiting for hours - killed 126. He said the dead included 109 people from Foua and Kfarya, among them 80 children and 13 women.
No one has claimed the attack, but both the Islamic State group and the al-Qaida-affiliated Fatah al-Sham Front have targeted civilians in government areas in the past.
Anthony Lake, UNICEF's executive director, said in a statement Sunday that after six years of war and carnage in Syria "there comes a new horror that must break the heart of anyone who has one."
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