Sheik Sharif Sheik Ahmed, chairman of the Islamic Courts Union, said in a radio broadcast on Monday: "We want to restore peace and stability to Mogadishu. We are ready to meet and talk to anybody and any group for the interest of the people."
If the claim is confirmed it would be the first time control of the entire city had been wrested from Somalia's powerful warlords since they ousted the former president, Mohammed Siad Barre, in 1991.
Ali Nur, a warlord coalition militiaman, said members of the alliance were now fleeing. "We have no immediate plans. Most of our leaders have fled Mogadishu to Jowhar," he said.
About 350 people, mostly civilians, have been killed in street battles since February between the Islamist militia and a self-styled anti-terrorism coalition of warlords, the so-called Alliance for the Restoration of Peace and Counter-Terrorism (ARPCT).
Many analysts view the violence as a proxy war between the United States and Islamic militants.
The Islamic courts have also gained popularity by restoring a semblance of order to parts of the capital.
"We know that some of the warlords are receiving support from Washington in the form of payments that allow them to buy weapons," an analyst with the International Crisis Group (ICG) think-tank told Aljazeera.net.
"We know that some of the warlords are receiving support from Washington"
Suleiman Baldo, International Crisis Group
Suleiman Baldo, who heads ICG's Africa Programme, said Washington has neither confirmed nor denied that it supports the secular fighters.
"US officials say they are in partnership with people that help them in their war on terrorism," he said.
The US administration believes that some people linked to al-Qaeda are hiding in Somalia, including those who bombed the US embassies in Tanzania and Kenya in 1998.
It says the same al-Qaeda cell is responsible for the suicide bombing of an Israeli-owned hotel in Kenya in 2002 that killed 15 people and a simultaneous attempt to shoot down an Israeli airliner.
Sean McCormack, the US state department spokesman, said on Monday: "We do have real concerns about the presence of foreign terrorists in Somalia, and that informs an important aspect of our policy with regard to Somalia."
But Baldo said that the US alliance with the warlords had backfired. "The courts are getting stronger, some residents actually support them because they've restored a semblance of order and now they've captured Mogadishu," he said.
"The US is interested in immediate results and has failed."
He also said that Washington had not hesitated to shift alliances by supporting some of the very warlords that killed 18 US troops in 1993 in Mogadishu.
The battle was made famous by the film Black Hawk Down.
Residents said the Islamist fighters took the last warlord stronghold in the capital on Monday without a fight. Many expressed relief.
"The city is calm and we hope it will stay that way"
Samira Jama, resident
Mogadishu's Dayniile area was a stronghold of Mohamed Qanyare, a warlord who left two days ago after local elders ordered him out to stop battles that have caused heavy civilian casualties.
Samira Jama, a 19-year-old resident, said: "The city is calm and we hope it will stay that way."
Ali Abdikadir, another resident, said: "The Islamic courts announced they are in control of Mogadishu. They said they would work with residents to improve security in the capital.
"This is good news for us because the warlords were always engaged in battles, we are looking forward to a life without fighting."
Abdirahman Dinari, spokesman for Somalia's interim government, said he hoped for talks soon with the Islamist side.
"I hope the Islamic courts will realise the importance of the government and talk to us," he said. "We are ready to talk to them in order to prevent further suffering of the innocent Somali people."
The government, based in the southern city of Baidoa, 240km west of Mogadishu, has been too weak to stop the fighting or enter the capital.
"We are ready to talk to [the Islamic courts] to prevent further suffering of the innocent Somali people"
Interim government spokesman
Fearing more fighting, the UN pulled staff out of Jowhar, which could be the next target for the Islamists.
Before seizing the capital, the Islamist militia took control of the strategic town of Balad in a battle on Sunday that left 18 people killed.
The town controls the supply route from the warlord stronghold of Jowhar further north.
The interim government late on Sunday sacked four warlord ministers, including Qanyare, who was in charge of internal security, for their role in the fighting.