If the claim is confirmed, it will be the first time control of all the city has been wrested from Somalia's powerful warlords since they toppled Somalia's last effective government 15 years ago, that of former dictator Mohamed Siad Barre.
About 350 people, mainly civilians, have been killed in street battles since February.
After capturing nearly all of the lawless capital and a key warlord supply line on its northern outskirts at the weekend, the Islamists said they were cementing their authority over areas formerly held by the alliance.
At least 18 people died in the latest battle on Sunday, the Islamic militia took control of the strategic town of Balad, which controls the supply line from the warlord stronghold of Jowhar further north.
Witnesses said fighters loyal to the warlords have not been seen on the streets and many of their top commanders are said to have left Mogadishu amid signs any short-term attempt to re-take the city would be futile, according to witnesses.
"The era of warlords in Somalia is over," said resident Mohamed Asser, according to Reuters. "This morning Mogadishu is under only one hand, the Islamic courts."
In a statement read over local radio stations, the chairman of the city's Joint Islamic Courts, Sheikh Sharif Sheikh Ahmed, said 15 years of warlord control of Mogadishu was over and urged residents to accept the new leadership.
"The Joint Islamic Courts are not interested in a continuation of hostilities and will fully implement peace and security after the change has been made by the victory of the people with the support of Allah," he said.
"The JIC will take care of the safety of the people and freedom of individuals and will eradicate any sort of hostilities brought about by inter-clan fighting," Mr Ahmed said.
Senior court officials said preparations are being made to bring all of Mogadishu under Islamic law.
"The creation of new Islamic courts in the areas that were controlled by the warlords is expected very soon," one official told AFP on condition of anonymity.
The interim government, based in the southern city of Baidoa, has been too weak to stop the fighting or enter the capital.
Late on Sunday it sacked four warlord ministers, including Mohamed Afrah Qanyare, who was in charge of internal security, and Muse Sudi Yalahow, the commerce minister, for their roles in the fighting.
The other ministers were Religious Affairs Minister Omar Mohamed Mohamud and Militia Disarmament Minister Bootan Isse Alim. Along with Muse Sudi, these two are said to be heading to neighbouring Kenya.
Interim Prime Minister Ali Mohammed Ghedi said his government wants to begin dialogue with the group, according to the BBC. Mogadishu residents have reportedly given a cautious welcome to the news.
"The Islamic courts announced they are in control of Mogadishu. They said they would work with residents to improve security in the capital," resident Ali Abdikadir said by telephone after attending a public meeting, according to Reuters.
"This is good news for us because the warlords were always engaged in battles. We are looking forward to a life without fighting," he added.
"The city is calm and we hope it will stay that way," 19-year-old resident Samira Jama said.
Warlords US-backed: claims
The warlords were operating under the Alliance for the Restoration of Peace and Counter-Terrorism (ARPCT) since February.
The Alliance was reportedly created with US support in a bid to curb the growing influence of Mogadishu's 11 Islamic courts and hunt down extremists, including Al-Qaeda members they are accused of harbouring.
Washington has never publicly confirmed or denied its support for the alliance. But US officials told AFP they had given the warlords money and intelligence help to rein in "creeping Talibanisation" in Somalia.
The Horn of Africa nation was plunged into anarchy with the 1991 ouster of strongman Mohamed Siad Barre, and analysts have long warned it could become a hotbed for radical Islam along the lines of Afghanistan.