It said security forces intercepted three suicide car bombers on Sunday morning, blowing up two at the entrance to the city.
The third was able to enter the city and blew himself up after being surrounded in Tahrir Square, causing several deaths and leaving a number of people wounded, Al Jazeera reported.
Syrian state news agency SANA quoted a senior police official saying that "terrorists simultaneously blew up three cars", two of them on the road to Damascus airport southeast of the capital and a third in the eastern Sahat Al-Tahrir district.
"The terrorist bombings killed and wounded several civilians and caused physical damage to the area," the official said.
It was not immediately clear if the two explosions on airport road, at the entrance of the city, were caused by the suicide bombers or if their bombs were detonated by security forces.
Damascus has been spared the large-scale battles that have devastated other major Syrian cities during the country's six-year civil war.
But dozens of people have been killed in bombings, particularly on the outskirts of the capital.
In mid-March, bomb attacks on a courthouse and restaurant in Damascus killed 32 people. They were claimed by the Daesh (also known as ISIL or ISIS) group.
That came days after two explosions that left 74 dead in the capital's Old City and were claimed by the hardline Hay'at Tahrir al-Sham alliance, led by the former al-Qaeda affiliate Jabhat Fateh al-Sham.