The African Union says its peacekeeping troops are now in control of more than 40% of the Somali capital, Mogadishu.
The AU envoy to Somalia, Wafula Wamunyinyi, told the BBC that the force backing the government had moved into several new positions in recent days.
Meanwhile deep splits have been reported in the main Islamist group.
But correspondents say that even though al-Shabab has been weakened by the divisions, it is unlikely that AU forces control so much of the city.
The 6,000-strong African Union force, deployed to back the weak UN-backed interim government, previously controlled only a few areas around the presidential palace, the harbour and the airport.
But Mr Wamunyinyi said the AU had cut off Islamist insurgents from one of their key bases in the city's main Bakara market.
The Islamists have used the busy market as a hiding place, shielding themselves amongst civilian traders, and firing rockets at government and AU positions.
BBC Africa analyst Mary Harper says a large number of Somali militiamen have reportedly distanced themselves from al-Shabab's de facto leader, Ahmed Godane, who has a significant following of foreign fighters.
If he is sidelined, the more radical wing of the movement - which is closely aligned to al-Qaeda - may have to move itself and its foreign supporters outside Somalia, she says.
"They [the Islamists] are at their weakest. If we had sufficient troop numbers we could move quickly," Mr Wamunyinyi said at press briefing in neighbouring Nairobi, Reuters news agency reports.
The AU has appealed to the international community for more funds to increase the force to 20,000 and asked its members to consider sending troops to support the Ugandan and Burundian contingents in Mogadishu.
But at a separate briefing in Cape Town, South Africa's defence secretary said her government was not going to send soldiers to Mogadishu.
"The answer to Somalia is a simple 'No'," Mpumi Mpofu said.
Somalia has been wracked by conflict ever since President Siad Barre was ousted in 1991.