The blasts on Wednesday occurred just days after a bomb attack killed two women election workers there.
It was not immediately clear if there had been any fatalities in the blasts, which the Pakistan-based Afghan Islamic Press news agency said hit two security posts.
At least three of those wounded were police officers and five were children.
"This is the work of terrorists," said security official Abd al-Rahman.
The blasts came just days after a bomb killed two women working to register female voters for the UN-Afghan electoral body in the same city on Saturday.
A day earlier, suspected Taliban fighters killed 16 people in the southern province of Zabul after finding they had voter registration cards.
Threat to elections
The 20,000-strong US-led force pursuing Taliban and their supporters has warned of stepped-up violence ahead of the elections.
Food convoy of US food was
Meanwhile, a convoy of trucks carrying food for US forces in a southern province was attacked and 12 Afghan drivers and workers are declared missing, a provincial military commander said on Wednesday.
The fighters attacked the convoy and burned four trucks at about midday on Tuesday on a road about 80km (50 miles) north of the city of Kandahar, Kandahar province corps commander General Khan Mohammad said.
"They took 12 truck drivers and workers with them," he said.
The trucks had been transporting food from Pakistan for US forces in Afghanistan, said Khalid Pashtun, the spokesman for Kandahar province.
Afghanistan has suffered a wave of conflict in recent months blamed on fighters who have vowed to disrupt preparations for elections due to be held in September.