Labour will seek to force Mrs May into a decisive second Commons showdown on her Brexit deal by February 26.
Prime Minister Theresa May will pledge this week to give parliament another chance to voice their opinions on Brexit by February 27 as she tries to buy more time to negotiate a new deal with the European Union.
May will promise parliament another vote on other Brexit options if a deal is still not ready by the end of February.
But there is no commitment to hold a binding vote on the deal itself by the end of the month.
"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".
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.
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.
Labour leader Jeremy Corbyn has written to the Prime Minister setting out five demands that would have to be met for his MPs to support a deal, including a permanent customs union and close alignment with the single market.
Communities Secretary James Brokenshire said: "I think that gives that sense of timetable, clarity and goal on what we are doing with the European Union - taking that work forward and our determination to get a deal - but equally knowing that role that Parliament very firmly has".
The move led to a backlash from pro-EU Labour MPs, but Mr Starmer defended the approach and warned against a split in the party.
Treasury Chief Secretary Liz Truss refused to rule out quitting if Mrs May did accept the demand for a customs union.
Housing minister James Brokenshire said on Sunday May would commit to giving parliament another debate on Brexit with the chance to vote on alternative options, if a deal had not yet been agreed and voted upon by then.
May will ask lawmakers on Thursday to reaffirm that they support her bid to renegotiate the backstop, a government source said.