Klestil's death came fewer than two days before he was due to leave his largely ceremonial post after two six-year terms in office, the BBC reported.
Social Democrat Heinz Fischer, who was elected in April, will become the new president.
After a successful career in diplomacy, Klestil in 1992 replaced Kurt Waldheim as Austria's president. Waldheim was found to have worked as an officer in a Nazi unit that committed war crimes.
Klestil is credited with improving Austria's image abroad, which had been marred by the scandal surrounding Waldheim. Klestil expressed sympathy for the victims of the Holocaust during a speech to the Israeli parliament in 1994.