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The ex-prime minister argues that MPs must vote with their "own conscience" on whether the deal on offer will leave the UK better or worse off.
And, in a speech in London, he said they could decide on a new referendum.
Leading Tory backbencher and Brexit backer, Jacob Rees-Mogg, said Sir John had been wrong on Europe in the past "and he is getting it wrong again".
"We had a democratic vote, the decision has been taken and what he is trying to do is overturn that," he told BBC News.
Speaking to the BBC's political editor Laura Kuenssberg, Sir John insisted he was not "targeting" Theresa May and he genuinely wished her well in getting the best deal for Britain.
But he said some of the advice she was getting from within her own party was "very unwelcome from the point of view of the wellbeing of the country". [...]