Mr Blair says introducing tougher immigration policies would fulfil the will of Brexit voters so make it unnecessary for the UK to leave the bloc.
The institute he founded after he left Downing Street in 2007 has written a policy document that says Brexit is not “the sole, or even the best way to” overcome “public anxiety about immigration”.
It says the Government should “seek to negotiate a strengthened ’emergency brake'” to put in place “temporary controls on free movement” which would allow “control over immigration” while leaving open the option of “remaining within the EU”.
But former Chancellor Ken Clarke told Sky News such a plan was “hopeless”, while Conservative minister Dominic Raab said his idea had been tried already and been unsuccessful.
Mr Blair told BBC’s Andrew Marr show: “I think you could argue in Europe today for an emergency brake that didn’t just deal with benefits, but also dealt with the influx of people and I think there are things we could do domestically like registering people that come in and those that go out.
“This is post the Brexit vote. At the moment, you either do leave, or we have the status quo. Brexit is a distraction not a solution to the problems this country faces.
“If members of parliament really believe that, then their obligation is to set out solutions and deal with the actual problems people have and not do Brexit which is going to distract us from those solutions.”
But the Government’s justice minister Dominic Raab said: “Tony Blair was the guy that introduced the open door immigration we are now trying to resolve because the public feel very concerned about it.
“We’ve also tried to get the kind of deal that he talked about with the EU and the EU said no. And then there is the small matter of the referendum when the British people decided to leave the EU and we are going to deliver that.”
Ken Clarke, who was in the Cabinet until 2012 and was against Brexit before the referendum last year, said: “Tony still thinks we can stay in the EU. I think the mood in the country… It’s hopeless to expect that.
“What we now need to address is the practical consequences, what’s our new relationship. The single market and the customs union simply mean we don’t start putting new barriers between ourselves and our biggest single market. That, we probably can achieve.”
Defence Secretary Michael Fallon told Marr: “It’s a bit late now this epiphany, I’m not sure where he’s been – well we know where he’s been, he’s being travelling the world.
He added: “The country has decided we’re leaving the European Union, we’ve got to get on with that.”
Labour also said it is committed to Brexit.
Shadow health secretary Jon Ashworth told Sky News: “Brexit is going to go ahead. The British people voted for Brexit, but we are trying to do is get a deal that safeguards jobs and the economy.”
The criticism of Mr Blair comes as Parliament prepares to vote on the first reading of the EU Withdrawal bill, which will move thousands of pieces of EU law into domestic legislation.