Tony Blair has defended his call for new controls on EU migration as a cabinet minister accused him of a belated "epiphany" on the issue.
The ex-PM said the UK could stay in the EU after all with new curbs in place.
He claimed this would address people's "grievances" without the "sledgehammer" of Brexit.
Critics have pointed to his Labour government's decision not to apply transitional controls to eastern European migrants in 2004.
Mr Blair's proposals are to "tighten" existing free movement rules, including on benefit entitlement, and seek to negotiate an "emergency brake" on EU migration in certain sectors.
Speaking on the BBC's Andrew Marr show, the former prime minister - one of the most prominent anti-Brexit campaigners - said he accepted last year's Leave vote but that there were ways of controlling EU immigration without leaving.
"Brexit is a distraction, not a solution, to the problems this country is facing," he said.
Mr Blair said he believed Brexit would go ahead "unless it starts to become obvious that the public is having second thoughts" - and that "hasn't become obvious yet".
"If we put this case to people, maybe they will listen. If they don't - I accept it goes forward," he said.
Unlike France and Germany, which did not give migrants from the 10 countries which joined the EU in May 2004 full access to their labour market until 2011, the then-Labour government did not insist on any transitional controls.
Mr Blair said: "The situation back then was different."
He told the Marr show the economy had been strong when he left office in 2007 before the financial crash, adding: "You've got to listen to what people are saying and react to it."
"It's a bit late now, this epiphany", responded Defence Secretary Sir Michael Fallon on the Marr show.
Sir Michael said "election after election" had shown the public wanted "proper controls" over immigration.
"I think it's a pity he didn't think of that when all these new countries were admitted to the European Union on his watch," he added.
Pro-EU Conservative Ken Clarke told Sky News it was "hopeless" to think the UK could stay in the EU, given the "mood of the country".
The government has said free movement will end with the UK's withdrawal from the EU, and a forthcoming immigration bill will set out its plans in detail.
Last week, leaked draft plans suggested firms would have to recruit locally unless they could prove an "economic need" to employ EU citizens and ending the right to settle in the UK for most European migrants.
The report by Mr Blair's Institute for Global Change suggests reforming free movement rules, without "excessive" restrictions which would jeopardise the UK's membership of the EU single market.
It says the "structure of free movement" could be kept "broadly intact" with rules on access to benefits tightened.
Under existing rules, citizens of other EU countries can be removed after six months if they have not found a job, have no realistic possibility of finding one, and require support from the welfare system.
Mr Blair's report says EU nationals should already have an offer of work when they arrive and says those who did not earn permission to stay would be banned from opening a bank account, renting a home or claiming benefits.
'Out of touch'
The report also proposes restricting free healthcare for unemployed migrants and letting universities charge EU nationals higher tuition fees than UK students.
It says an "emergency brake" could be negotiated with the EU, which would allow the UK to impose temporary restrictions in certain sectors when migrant numbers were high.
An "emergency brake" - which would have affected migrants' benefits - was key to the package of EU reforms ex-PM David Cameron tried to use to persuade voters to back staying in the EU last year.
The co-chairman of the campaign group Leave Means Leave, Richard Tice, said Tony Blair's attempt to deny democracy would be seen for what it was, and ex-UKIP leader Nigel Farage, referring to immigration, said Mr Blair was "the reason we are in this mess".
Unite union leader Len McCluskey said Mr Blair was "as out of touch now as he was in 2004".
He told Radio 5 live's Pienaar's Politics that "greedy bosses", not migrant workers, were to blame for undercutting wages, saying that unionisation and regulation were the only way to prevent this from happening.
Opening the TUC's annual conference in Brighton, general secretary Frances O'Grady said the government had "no realistic negotiating strategy" and was taking the UK towards a "kamikaze Brexit".