The former Prime Minister waded into the Brexit debate to warn of the “terrible public backlash” Mrs May is risking if the UK is left poorer and weaker after leaving the EU.
He said the Government should make the “brave” decision to offer the free vote to “let Parliament decide, or put the issue back to the British people”.
Sir John said MPs should be given the power to accept or reject the final deal, send negotiators back for improvements or order a second referendum for the public to decide.
He added that the public was realising it had been misled and had “every right to reconsider the decision”.
Speaking in London just two days before Mrs May sets out her own vision for a post-Brexit Britain, Sir John said that the close referendum result meant the government should negotiate a Brexit deal but not pull the UK out of the EU “at any cost”.
The former Conservative leader, who served as Prime Minister in the nineties, said the “meaningful vote” that had been promised to MPs on the final Brexit deal should be “decisive”.
He said parliament should have the power to accept or reject the final outcome, send negotiators back to demand improvements or order another referendum for the public to approve the decision.
“That is what parliamentary sovereignty really means,” he said.
Sir John called on Mrs May to stand up to the hardline Brexiteers in her party and drop her “red lines” of taking Britain out of the single market and customs union which he said were unrealistic and were “boxing the Government into a corner”.
He accused the Tories of not understanding businesses concerns over Brexit and said the only reason the party had not “haemorrhage[d] business support” was due to fears of Labour’s Jeremy Corbyn taking power.
Mrs May’s current position was “tilted to ultra-Brexit opinion”, even though hardline leavers had so far been wrong in nearly every promise they made to referendum voters, he said.
“Many electors know they were misled: many more are beginning to realise it. So, the electorate has every right to reconsider their decision.”
He said the slim referendum majority did not give the Government an “overwhelming mandate to ignore the reservations of 16 million voters who believe it will be a harmful change of direction for our country”.
“Of course, the ‘will of the people’ can’t be ignored, but Parliament has a duty also to consider the well-being of the people,” he added.
“No-one voted for higher prices and poorer public services, but that is what they may get. The emerging evidence suggests Brexit will hurt most those who have least. Neither Parliament nor Government wish to see that.”