Dagan, who battled liver cancer and had undergone a transplant, led the Mossad from 2002 through 2010.
He was reportedly tasked with sabotaging the nuclear programme of Israel's arch-foe Iran to prevent it from developing atomic weapons.
But while leading that secret war, he also strongly opposed a military strike against Iran, a position shared by then military chief of staff Gabi Ashkenazi.
Prime Minister Benjamin Netanyahu and then defence minister Ehud Barak were reported to have given the order in 2010 for the military to prepare such a strike, which was never carried out.
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Under Dagan's leadership, the Mossad is believed to have assassinated Iranian nuclear scientists, caused explosions at nuclear facilities and used computer viruses to damage uranium centrifuges.
The Mossad has never confirmed such operations.
Several other controversial operations were attributed to the Mossad during Dagan's unusually long tenure, including a 2008 car bomb in Damascus that killed top Hezbollah operative Imad Mughniyeh.
Others included a 2008 air raid in Sudan against an alleged Iranian arms convoy said to be destined for Palestinian Islamist movement Hamas and a 2007 bombing of a suspected desert nuclear site in Syria.