Some EU leaders may be prepared to compromise on the free movement of people to help Britain stay in the single market, Tony Blair has said.
He told the Today programme one option was for Britain "staying within a reformed EU".
The ex-PM said he would not disclose conversations he had had in Europe - but insisted he was not speaking "on a whim".
The government insists Brexit will give the UK greater control of its borders.
Labour's shadow chancellor, John McDonnell, said Mr Blair "hadn't really listened to the nature of the debate going on in the pubs, the clubs and school gates".
"We have to respect the referendum result," Mr McDonnell said, adding that Labour could "negotiate access to the single market".
Mr Blair spoke to the BBC after he argued in an article for his own institute that there was room for compromise on free movement of people.
He told Today the situation in Europe was different to when Britain voted to leave the EU - a move Mr Blair described as "the most serious it's taken since the Second World War".
He said France's new president, Emmanuel Macron - whose political party was formed last year - was proposing "far-reaching reforms" for the EU.
"Europe itself is now looking at its own reform programme," Mr Blair said.
"They will have an inner circle in the EU that will be part of the eurozone and an outer circle."
When pressed on what evidence there was to suggest European nations would compromise, Mr Blair said: "I'm not going to disclose conversations I've had within Europe, but I'm not saying this literally on the basis of a whim.
"They will make reforms that I think will make it much more comfortable for Britain to fit itself in that outer circle."
He said "majorities" of people in France, Germany and the UK supported changes around benefits and with regards to those who come to Europe without a job.
"I'm not saying these could be negotiated," Mr Blair said.
"I'm simply saying if we were looking at this from the point of view of the interests of the country, one option within this negotiation would be Britain staying within a reformed European Union."
He said the majority of EU migrants in the UK are "people we want in this country".
EU leaders have previously said the UK must accept free movement of people if it wants to stay inside the single market.
But in his article for the Institute for Global Change, Mr Blair said senior figures had told him they were willing to consider changes to one of the key principles of the single market.
"The French and Germans share some of the British worries, notably around immigration, and would compromise on freedom of movement," he wrote.
But last week the EU's chief negotiator, Michel Barnier, said the freedom of movement of people, goods, services and capital - the key principles of the single market - were "indivisible".
Prime Minister Theresa May has pledged to control EU migration and has reiterated her commitment to reducing net migration to the tens of thousands.
She has said that outside the single market, and without rules on freedom of movement, the UK will be able to make its own decisions on immigration.
Mr Blair also said more was known now about the effects of the Brexit process on the UK.
"We know our currency is down significantly, that's a prediction by the international markets as to our future prosperity. We know businesses are already moving jobs out of the country.
"We know last year we were the fastest-growing economy in the G7. We're now the slowest."
Mr Blair accepted Labour was behind its leader Jeremy Corbyn "for now".
But he warned if Brexit was combined with leaving the single market, and "the largest spending programme Labour had ever proposed" the country "would be in a very serious situation."
Mr Blair said leaving the single market was a "damaging position" shared by Labour and he urged the party's leadership to champion a "radically distinct" position on Europe.
But Jeremy Corbyn said Labour's position on free movement was "very clear", adding: "We would protect EU nationals' rights to remain here, including the rights of family reunion."
Responding to Mr Blair's comments, the party leader said: "I think our economy will do very well under a Labour government.
"It will be an investment-led economy that works for all - so we won't have zero-hour contracts, insecure employment.
"We won't have communities being left behind."
'Out of touch'
Mr Blair has previously said Brexit was an issue he felt so strongly about, that it tempted him to return to politics.
But Labour MP Frank Field, who backed Brexit, said he did not think Mr Blair was "a person to influence public opinion now".
"We're now set on the course of leaving [the EU]. We actually need a safe harbour to continue those negotiations when we're out.
"And I wouldn't actually be believing those people who are set on destroying our attempts to leave, who are now appearing as wolves in sheep's clothing."
Richard Tice, of pro-Brexit group Leave Means Leave, said Mr Blair's comments "demonstrate how out of touch he is with British voters".
"The former prime minister believes that freedom of movement is the only issue with the EU, when in reality the British people also voted to leave in order to take back control of our laws and money and no longer be dictated to by the European Court of Justice," he added.
Conservative MEP David Campbell Bannerman said Mr Blair's assertion that Britain could find a way to remain within a reformed EU was a "dodgy claim, as opposed to a dodgy dossier".
"We've heard this all before. David Cameron was given such assurances and in the end the EU did nothing for him.
"If they do nothing for Cameron, they're not going to do anything for Blair, I'm afraid."