Blair repeated his claim to reporters that Brexit will be economically damaging - something backed up by the government's own assessment and that of the Bank of England - and said May should have been more transparent about trade-offs when she set out to negotiate with Brussels.
DUP leader Arlene Foster dismissed it as "legislative tinkering" while Tory Brexiteers said it was "desperate".
Supporters of a clean break with the European Union say the backstop, meant to ensure no hard border between British-ruled Northern Ireland and the EU-member Irish Republic, could leave Britain forced to accept European Union regulations indefinitely, or Northern Ireland treated differently from the rest of Britain.
The EU's legal team argued that allowing the United Kingdom to unilaterally withdraw Article 50 would be a "disaster" for the European project because other countries could exploit the precedent in a bid to win better membership terms without having a real intention to leave.
Dr Fox replied: "The Government will continue to make the case for what it believes is a balanced and reasonable agreement, but of course the Government will want to talk to members, it will want to look to ways to give a reassurance to the House wherever we're able to do that".
"If it can't, then I certainly would welcome the vote being deferred until such time as we can answer that question".
Under the current deal the "backstop" would come into effect at the end of the Brexit transition period, which is due to last until the end of 2020 but may be extended.
It's not surprising that MPs would have a vote on that. Yes I am meeting colleagues, I'm listening to colleagues' concerns.
Brexiteers and Northern Ireland's Democratic Unionist Party, which props up May's government, worry that the proposed deal could leave Britain under European Union regulations indefinitely, or Northern Ireland could be treated differently to the rest of the UK.
May also took aim at those in Parliament that she said were trying to "frustrate Brexit".
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For example, it includes initials that do not match those of any immigration service centers that issue the cards. Trump boasted during the presidential campaign that no illegal immigrants worked at any of his properties.
But Conservative Brexiteer Steve Baker said: "Giving Parliament the choice between the devil and the deep blue sea is desperate and will persuade very few".
The PM's deal is now supported by only 27 per cent of Brits putting her offer level with a no deal agreement (even though a no deal has a lead support in 30 seats).
DUP leader Arlene Foster, whose party opposes Mrs May's deal, warned that the amendment would not be enough, tweeting: "Domestic legislative tinkering won't cut it".
"On Monday the Attorney General told MPs that it would be "contrary to the national interest" to release his legal advice of November 13", said Ms Lucas.
Media captionConfused by Brexit jargon?
Mrs May faced calls to postpone Tuesday's vote, with senior Conservative MP Sir Graham Brady, chairman of the backbench 1922 Committee, saying he would welcome the vote being deferred if no solution could be found to differences within the party over the backstop.
The chancellor will visit a school in Chertsey, Surrey, while the health secretary will go to a hospital in Portsmouth on Friday.
Around 30 ministers in total will join the push to gain support for the withdrawal agreement.
"I want to see central to any agreement that we can control our own borders and that we can control our trade policies".