"Operation Indian Ocean started late last night.... The enemy is fleeing and the forces are making successful advances so far," said Abdukadir Mohamed Nur, the governor of southern Somalia's Lower Shabelle region.
Witnesses reported hearing the booms of heavy shelling, as well as seeing convoys of tanks and armoured vehicles heading towards the Shebab stronghold of Bulomarer, some 160 kilometres southwest of the capital Mogadishu.
The Al-Qaeda-linked Shebab are still a powerful and dangerous force but have lost a string of towns to the 22,000-strong AU mission in Somalia (AMISOM).
"We are heading towards Bulomarer town," Nur added. "The operations will not stop until the Al-Qaeda militants are eliminated."
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In January 2013, French commandos carried out a botched raid on Bulomarer in a bid to free a secret agent held hostage.
The bid failed and resulted in the death of two French soldiers and the hostage.
"We are hearing heavy shelling," said Ali Mohamed, who lives close to the Bulomarer, adding that fighting close to the town.
"I saw an AMISOM military convoy including several tanks this morning headed towards Bulomarer and Barawe," said Hussein Mumin, another local resident.
The Shebab are fighting to topple Somalia's internationally-backed government, and regularly launch attacks against state targets, as well as in neighbouring countries that contribute to the AU force.
The fighting comes as UN and aid workers warn large areas of Somalia are struggling with dire hunger and drought, three years after famine killed more than a quarter of a million people.