Turabi was taken to the intensive care unit of Khartoum's Royal Care hospital "after suffering a heart attack in the morning and died" there, the source told AFP.
As news of his death became known, state television interrupted its regular programming and broadcast Islamic verses from the Koran that are recited for the dead.
A key figure in Bashir's regime for a decade after his 1989 coup, Turabi later became one of its fiercest critics and led the opposition in urging a Tunisia-style uprising.
Turabi was the only Sudanese politician to support a warrant issued for Bashir's arrest by the International Criminal Court (ICC) on charges of war crimes, crimes against humanity and genocide over the regime's conduct of the conflict in Darfur.
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After breaking ranks with Bashir he formed his own party, the Popular Congress Party.
Turabi was detained several times over a career spanning four decades, including in January 2009 two days after he urged Bashir to surrender to the ICC.
The Western-educated Turabi held a master's degree in law from London and a doctorate from Sorbonne University in Paris.
He spoke English, French and German fluently as well as Arabic, and his language skills helped him gain access to foreign news media through which he issued repeated calls for an international Islamic revolution.