But prospects for the resumption of talks, the third round of which collapsed last week in Sudan, remained unclear as the two sides continued to prepare for a battle.
Islamist conditions for meeting face-to-face with the government appeared not to be addressed and there was no immediate response from the government, which had sought a delay in the parliament speaker's visit to Mogadishu.
Preparations for war continued outside the government seat of Baidoa, where residents reported hearing heavy gunfire, even as the Islamists welcomed speaker Sharif Hassan Sheikh Aden's initiative.
"The talks (with Mr Aden) were so critical and the Islamic courts have accepted the offer from the speaker for talks," said Sheikh Mohamed Ibrahim, a senior official in the Supreme Islamic Council of Somalia (SICS).
"Now, we are ready to continue the peace process in Khartoum," he told news agency AFP.
However, there was no response from the government, which yesterday had urged the influential Sheikh Aden to postpone his unauthorized mission until he met with the cabinet to coordinate strategy.
Mr Ibrahim said the Islamists would meet on Tuesday to work out specifics for the resumption of talks.
The Islamists seized Mogadishu in June after months of fierce fighting and have since taken control of most of southern and central where they have imposed strict Sharia law, leading to fears of a Taliban-style takeover.
Mr Aden was warmly received in the capital by senior Islamist officials, including SICS supreme leader Sheikh Hassan Dahir Aweys, a hardline cleric designated a "terrorist" by the United States for alleged al-Qaeda ties.
"We don't refuse negotiation and we welcome what the speaker has offered us," Sheikh Aweys told reporters at a welcoming ceremony after Mr Aden's arrival.
"I will avert an imminent war in Somalia," Mr Aden predicted at the ceremony.
Sheikh Aden was not immediately available to comment after the meeting, but Mr Ibrahim said he had suggested moving the government from Baidoa, about 250 kilometres northeast of Mogadishu, to the capital.
A move to the capital from Baidoa, where the administration is based due to security concerns in Mogadishu, is likely to be opposed by government officials who accuse the Islamists of being behind a failed September suicide car bomb attempt to assassinate interim President Abdullahi Yusuf Ahmed.
Somalia has been without a functioning central authority since the 1991 ousting of strongman Mohamed Siad Barre.