According to state television, the death toll included both civilians and members of the security forces and blamed “armed terrorist gangs” for the blasts.
There are conflicting reports about whether the rebel Free Syrian Army (FSA) was behind the attacks.
A spokesman appeared to confirm that it was, but another blamed the government.
Col Malik al-Kurdi, the FSA’s deputy leader, said it had been monitoring the activity of security forces personnel and members of the pro-government Shabiha militia inside a Military Intelligence compound and a riot police base in Aleppo on Friday morning.
“When they were gathering in a square to go to the mosques and repress demonstrations, two groups from the FSA targeted the two buildings with small arms and rocket-propelled grenade fire,” he said.
“After violent clashes, there was an explosion inside the Military Intelligence building. At first we didn’t know what it was, but we think it was the regime trying to stop the operation of the FSA,” he added.
Another FSA spokesman, Col Mahir Nouaimi, told AFP: “This criminal regime is killing our children in Homs and carrying out bomb attacks in Aleppo to steer attention away from what it is doing in Homs.”
Opposition activists also said the government was responsible, accusing it of trying to discredit the uprising. Suspicious activity by security personnel had been noticed shortly before the blasts, they added.
Syrian state TV broadcast images showing at least five corpses and mangled body parts.
A weeping TV reporter said the bomb targeting an intelligence building went off near a park, where people had gathered for breakfast and children had been playing.
Some children were killed in the blast, he said, holding up an inline-skate.
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