According to media reports, Blair said European Union leaders would be willing to change their rules on immigration and free movement to enable Britain to stay in the 28-nation trading and economic bloc.
Mr Blair suggested the "will of the people" may be changing as the difficulties of Brexit negotiations become apparent and called for a "proper debate" over the different options, including remaining in a reformed EU.
Pro-EU voters will eventually begin to understand Labour holds the "same damaging position" as the Tories, to leaving the single market to end free movement, and the party should instead champion a "radically distinct" position on Europe.
Asked about Mr Blair's comments that Brexit followed by a Jeremy Corbyn government would leave Britain flat on its back, Mr McDonnell said: "If you listen to ordinary people in this country now, what they want is basically a Brexit that will protect their jobs and protect the economy".
"This is a completely changed situation in Europe". The members of the Euro zone will integrate economic decision-making.
President Macron's election as French President has changed the "political dynamics of Europe" and reform is "now on Europe's agenda", Mr Blair says.
"Inevitably, therefore, Europe will comprise an inner and outer circle".
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"The rational examination of the options would include wisely the one to negotiate with the United Kingdom, maintaining it inside of a Europe ready to reform and make it half of the way towards us", wrote the former Prime minister.
"European leaders, after discussions that I have had, are willing to consider changes to go in the direction of the United Kingdom, including the freedom of movement".
"Yet this option is excluded".
German and French voters would favour similar controls, the polling suggested.
"Our savings ratio is the lowest for 50 years, the investment community internationally has now gone really negative on us, the currency's down 10, 12 per cent, investment in the motorcar industry for example is down 30 per cent, living standards are stagnating, I mean this is causing us real damage".
"On any basis, leaving is complex and will take years".
"I think it's absolutely necessary that it doesn't happen because I think every day is bringing us fresh evidence that it's doing us damage economically", Blair said, in a preview of Sky News' Sophy Ridge on Sunday programme.