The alliance of warlords has accused the Islamists of indiscriminate shelling, designed to provoke fear among the civilian population.
The fighting between the Islamists and a group of warlords has killed some 120 people, in the worst clashes for years.
Many Somalis accuse the US of backing the alliance of warlords.
The BBC's Mohammed Olad Hassan in Mogadishu says this belief is fuelling the violence.
A United Nations team says the Islamist alliance had gained ground since the fighting began last Sunday. Our reporter says several new parts of Mogadishu have come under attack, but the front line of the main battle has hardly moved from the northern suburbs where the fighting started.
Most of those killed in the fighting are civilians caught in cross-fire or hit by stray bullets or shells, he says.
The rival groups used heavy artillery and mortar rounds, as the fighting raged overnight , with many buildings destroyed.
"There is very heavy fighting here with huge shelling," Mogadishu resident Abdinasir Mohamed told the AFP news agency.
"We don't know where to go, where is safe," said shopkeeper Maryan Ibrahim.
Arms embargo
United Nations envoy to Somalia Francois Fall told the BBC's Focus on Africa programme that the UN was trying to negotiate a ceasefire between the rivals groups.
The US has not confirmed or denied backing the warlords but says it would "work with responsible individuals... in fighting terror".
A report by a UN committee on Wednesday warned that an unnamed country is flouting the arms embargo on Somalia.
The UN Security Council rejected the committee's recommendations for targeted sanctions and tighter controls on the illegal flow of weapons into Somalia, where guns are openly sold in markets.
This is the second round of Mogadishu's most serious fighting in a decade. In March, clashes between the two sides killed at least 90 people.
Lawless country
The militia linked to the Joint Islamic Courts have restored order to some parts of the city by providing justice under Sharia - Islamic law.
The alliance of warlords earlier this year created the Alliance for the Restoration of Peace and Counter-Terrorism.
It accuses the Islamic Courts of sheltering foreign al-Qaeda leaders, while the Courts say the alliance is a pawn of the United States.
Last week, Somali President Abdullahi Yusuf accused the US of funding the coalition of warlords.
The US has an anti-terror task force based in nearby Djibouti. Somalia has not had an effective national authority for 15 years since the ousting of President Siad Barre in 1991.
