The former prime minister will say in a speech later that people voted in the referendum "without knowledge of the true terms of Brexit".
He will say he wants to "build support for finding a way out from the present rush over the cliff's edge".
But former Cabinet minister Iain Duncan Smith said Mr Blair's comments were arrogant and utterly undemocratic.
Downing Street has said it is "absolutely committed" to seeing Brexit through.
Prime Minister Theresa May wants to trigger formal Brexit talks by the end of March - a move which was backed in the House of Commons by MPs last week.
Immigration issues
Mr Blair, who was UK prime minister between 1997 and 2007, will say in his speech to the pro-European campaign group Open Britain that those driving a withdrawal from the European Union "always wanted a hard Brexit".
"Indeed even the term 'Hard Brexit' requires amendment. The policy is now 'Brexit at any cost'," he will say.
"Our challenge is to expose relentlessly the actual cost, to show how this decision was based on imperfect knowledge which will now become informed knowledge, to calculate in 'easy to understand' ways how proceeding will cause real damage to the country and its citizens and to build support for finding a way out from the present rush over the cliff's edge."
Mr Blair, who campaigned to Remain in the EU, will say he accepts the verdict of June's referendum, but would recommend looking again at Brexit when "we have a clear sense of where we're going".
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