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.
"At least 14 people, mostly militia, died in the fighting this morning. The government's aim is to secure the city," Major Abdullahi Farah, a senior police officer, told Reuters.
"People were having illegal weapons and there are dozens of militia (men) in the city. This may enable al Shabaab to get cover."
Ahmed Hussein, a Mogadishu resident, said he was awakened by mortar rounds on Friday, and that al Shaabab fighters appeared to be moving back under the onslaught by the AU troops.
"The death toll is sure to rise," Hussein told Reuters from the Madina suburb which has residential homes, markets and schools.
Al Shabaab officials were not immediately available for comment.
Al Shabaab controlled large sections of Mogadishu until 2011 when African forces drove them out of Mogadishu and expelled them from most urban centers.
But the militants have since staged guerrilla attacks in parts of the capital.
Somalia has faced turmoil for more than two decades since the fall of dictator Mohamed Siad Barre tipped the country into chaos.
REUTERS