It found the Salafi-jihadist groups shared nearly identical ideologies and said challenging their interpretation of Islam was critical to defeating them.
"There's no point in just tackling the violence unless you tackle the ideology of extremism behind the violence," Blair said.
"You've got these broad ideological strands that lie behind a lot of this extremism. If you take, for example, some of the organizations in the Middle East, some of those clerics that are putting out the most extreme stuff -- they'll have Twitter followings that go into millions of people.
"These people are saying things about Jewish people -- about even those in their own religion who are different that we would regard as completely unacceptable -- and it's those waters of extremism in which the violent extremists can swim," he said.
"The majority of people within Islam do not support either the violence or the ideology. What we are talking about, however, is a radical Islamist way of thinking that results in extremism by small numbers of people, but that thinking is shared by larger numbers of people, and you've got to attack both -- the violence and the extremism, the thinking behind it," Blair said.
He said countries where extremism had taken hold needed to overhaul their education systems.
"We've got to use our negotiating power and might with these countries to say, 'You're going to have to reform the education systems that are educating millions of young people day in and day out to a view of the world that's narrow-minded, bigoted and hostile to those who are different.' "
Blair, who was Prime Minister from 1997 until 2007, led Britain when it went to war in Iraq. He later was the Mideast envoy for an international quartet of the United States, Russia, United Nations and the European Union to try to achieve a settlement of the Israeli-Palestinian conflict. He stepped down from that position
'Distorted religious principles'
The "Inside the Jihadi Mind" report authors said the propaganda of ISIS, al Qaeda and al-Nusra Front was built on shared "distorted religious principles" and produced "single-minded focus on violent jihad."
"When they attack one another, it is not ideological differences that drives the conflict, but differing narratives: the ways in which they apply their ideology to reality,"
the report said.
"The Salafi-jihadi movement will not be defeated by focusing on these narratives: it will only be defeated if we understand and engage the ideology."
Ed Husain, the foundation's senior adviser, said the groups "call for a caliphate of slavery, death and destruction."
"They justify their evil by abusively citing Scripture and creating religious certainty in the minds of angry, eager and obedient recruits," he said.
The center said the group's ideological foundations needed to be challenged by other interpretations of Islamic principles from mainstream Islamic theology.
"Jihadi groups thrive on simplicity: the more that they are forced to defend their interpretation of Islamic values, the harder it will be to maintain that simplicity," the report said.
Appealing to a younger generation
With 62% of Muslims under age 30, alternative ideologies must be provided in a way that "today's generation understand," the center said.
Celebrities, sports personalities and musicians need to be used in helping prevent the spread of extremism, it said.
"Counter-narratives must embrace the power of popular culture as well as the authority of religious voices to succeed. Meanwhile, many disenfranchised returning fighters have an important role to play in debunking the legitimacy of this ideology."
The center also recommended giving support to grass-roots Muslim responses and helping technology companies such as Twitter and YouTube promote "credible religious sharable material that rebuts the jihadi message."
"Search engines, social media, and video sharing websites could use algorithms and positive flagging systems to ensure this content is as visible as possible, as well as providing warnings that certain search terms might lead to extremist material."
Researchers gathered 114 online sources of propaganda for the report -- 48 from al Qaeda, 40 from ISIS and 26 from al-Nusra Front. They said only official sources for the groups were used.