"I counted eight bodies of dead insurgents in one place and three others in another location near the house of the commissioner," Mohamed Abdirisak, local resident, said.
Other people living in the area where the fighting took place gave the same death toll.
Daci said later that 17 opposition fighters had been killed by Somali government forces, but the claim could not be independently verified.
UN sanctions
The latest clashes came on a day the UN Security Council voted to impose economic sanctions on Somalis who contribute to violence and instability in the country.
The council unanimously adopted a British-sponsored resolution calling for an assets freeze and a foreign travel ban on Somalis who threaten peace talks, violate an arms embargo or interfere with aid deliveries.
"The prime goal of this is to provide a framework to stem the flow of arms into Somalia, which is causing such mayhem there," John Sawers, the British ambassador to the UN, said on Thursday.
The council is planning to set up a committee to establish a list of people who will face sanctions.
Piracy off the Horn of Africa coast has also become a major problem for regional shipping, underlining the international dimensions of Somalia's continued internal chaos and lawlessness.
Somalia has lacked an effective government since the 1991 removal of Mohamed Siad Barre, the former military ruler.
Advancing fighters
In recent weeks, an anti-government group, the Shabab, which is the former armed wing of the Islamic Courts Union, has seized or briefly occupied several towns on the outskirts of Mogadishu.
The group's fighters now control most of southern Somalia, including Marka, a strategic port 100km south of Mogadishu.
The Shabab aims to unite Somalia under sharia.
The group mounts daily attacks on government troops in Mogadishu, but analysts say it will be hard for them to take control of the city completely while heavily armed Ethiopian forces remain stationed there.