There was no immediate claim of responsibility for the bombings, the second wave of deadly attacks in the capital in less than a week after twin bombings killed 74 on Saturday.
But they came with the rebels fighting President Bashar al-Assad's regime increasingly divided and dispirited after a series of battlefield setbacks.
Negotiations to end the conflict have meanwhile made little progress, with rebels this week declining to attend negotiations in Kazakhstan.
Citing a police source, state news agency SANA put the initial death toll at 25 and said there were many wounded.
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An AFP correspondent at the scene in the Hamidiyeh neighbourhood said security forces had cordoned off the area and roads leading to it were blocked as ambulances and firefighters rushed to the building.
"We were terrified because the sound of the explosion was enormous," a lawyer in the building during the attack said.
The second blast hit not long after in the city's Rabweh area, an alert on state television said, but no further details were immediately available.
Damascus had already been reeling from Saturday's bombings, which mainly killed Iraqi pilgrims in the city to visit Shiite shrines.
That attack was claimed by former Al-Qaeda affiliate Fateh al-Sham Front, part of a rebel alliance that controls large parts of the northwestern Idlib province.
Rebel forces suffered a series of reversals during the sixth year of the war, including being forced from their onetime stronghold of east Aleppo in December.
The conflict began in 2011 with peaceful demonstrations inspired by similar movements during the so-called "Arab Spring", calling on Assad to implement reforms.
They started on March 15 after the arrest and torture of a group of students from the southern province of Daraa accused of writing anti-Assad graffiti.
The protests were put down violently, prompting demonstrators to pick up weapons and causing the uprising to spiral into an increasingly complex and brutal civil war that has also drawn in regional and international players.