At least three car bombs have exploded in the Syrian capital of Damascus after police chased the attackers through the city, state media said, killing at least 18 people and injuring more.

AFP said a monitoring group had announced the increased death toll, which had been put at seven in earlier reports.

Security forces stopped the bombers from reaching their targets which would have led to more deaths, officials told state TV, saying the bombers had aimed to hit busy areas on the first day back to work after the Eid al-Fitr holiday.

There was no immediate claim of responsibility. Damascus was hit by two separate, multiple suicide bomb attacks in March, one of them claimed by Islamic State and the other by the Islamist insurgent alliance Tahrir al-Sham.

The casualties in Sunday's attack occurred when one of bombers set off his device after being encircled near the Old City district of Bab Touma.

The other two car bombs were destroyed by the authorities, state media said.