The Sudanese government imposed a curfew on Saturday in response to the attack by Jem fighters. Al-Turabi denies that he has any links to the group.
Amr el-Kakhy, Al Jazeera's correspondent in Khartoum, said: "It's quite known that al-Turabi was a core partner with president Bashir in a coup that brought them to power in the 1980s."
"Since then, they have had their differences and al-Turabi was jailed. He was released on condition that he stay away from political life.
"Whenever something happens, al-Turabi is usually blamed - he is known for his criticism of the government."
State television said on Sunday that the curfew had been lifted in districts no longer affected by the fighting, identifying them as Khartoum, Khartoum North and the centre of Omdurman.
Sudan's government said on Saturday that Jem fighters had been "chased away".
The claim came after government forces and rebel fighters clashed in Omdurman on the western bank of the Nile opposite Khartoum.
But Khalil Ibrahim, the Jem's leader, said that his organisation was prepared to launch further attacks on Khartoum in an attempt to topple the Sudanese government.
"This is just the start of a process and the end is the termination of this regime…Don't expect just one more attack," he said on Monday.
Jan Eliasson, the UN envoy on Darfur, told Al Jazeera that the Jem has nothing to gain from the attack.
"I told representatives of Jem they will not achieve any sense of victory from this situation," he said.
"I fear the situation may get out of control, and this will prove disastrous to the people of Darfur."
Sudan broke off diplomatic relations with Chad on Sunday, accusing its neighbour of involvement in the previous day's attack.
"We are now cutting our diplomatic relations with this regime," Omar Hassan al-Bashir, the Sudanese president, said on state television.
"These forces [behind the attack] are all basically Chadian forces supported and prepared by Chad and they moved from Chad under the leadership of Khalil Ibrahim."
But the Chadian government denied "all involvement" .
"The government denies all involvement in this adventure which it condemns unreservedly, regardless of who was responsible," Mahamat Hissene, a government spokesman, said in a statement.
"The Chadian government, which has always supported efforts for peace in Sudan and in the region, encourages authorities and the opposition [Sudanese rebels] to maintain the voice of dialogue."
The Omdurman incursion was the first time that the conflict raging in western Sudan had reached the capital.
The violence in western Sudan has killed thousands and left millions displaced.