He said: "The Government will commit that if the meaningful vote, in other words the deal coming back, has not happened by 27 February then we would allow a further motion - votable in Parliament - to take place to give that sense of assurance as to the process moving forward as well in parallel".
Britain is due to leave the European Union on March 29th but Parliament has rejected May's divorce deal, leaving the Prime Minister to seek changes from a resistant EU.
But there is no commitment to hold a binding vote on the deal itself by the end of the month.More news: 28-Year-Old British Rapper Killed In Car Crash Enroute To Concert
"When you go through something like Brexit, it is very important that you keep the opposition strong and united", he said.
He told the Sunday Times he fears the Prime Minister is "pretending to make progress" but actually intends to return to Parliament after the March 21/22 European Council summit the week before Brexit and offer MPs a "binary choice" - her deal or no deal.
"It's this blinkered approach that's got us to where we are, with her never wanting to see where the real majority is in parliament". "There needs to be a day when Parliament says that's it, enough is enough".
Despite a series of setbacks for those campaigning for another Brexit referendum, Mr Blair said still hoped one might happen when people saw the "true alternatives" the country faces.More news: Liverpool reveals record profit of $162m
In his letter to Mrs May, the Labour leader set out five demands, including a permanent customs union and close alignment with the single market.
"It seems to me we're now at the point where we can have meaningful talks to get a deal between the main political party leaders", he said.
Treasury Chief Secretary Liz Truss refused to rule out quitting if Mrs May did accept the demand for a customs union.
"It would be economically very, very risky for Britain, and for the peace process in Ireland it would potentially be devastating, ' he told Sky News" Sophy Ridge On Sunday.More news: Danny Ainge confident Boston Celtics can keep Kyrie Irving
May will ask lawmakers on Thursday to reaffirm that they support her bid to renegotiate the backstop, a government source said.