Peacekeepers and Somali forces launched a campaign this month to weaken the al Shabaab militant group's violent hold on the Horn of Africa country.
Major Abdullahi Farah, a senior police officer, said: "At least 14 people, mostly militia, died in the fighting this morning. The government's aim is to secure the city. People have illegal weapons and there are dozens of militia in the city. This may enable al Shabaab to get cover."
Ahmed Hussein, a Mogadishu resident, said he was awakened by mortar rounds yesterday and that al Shaabab fighters appeared to be moving back under the onslaught by the AU troops.
Al Shabaab controlled large sections of Mogadishu until 2011, when African forces drove them out.