After the bombshell results, the premier vowed to 'redouble her efforts' to get concessions from Brussels but admitted it 'will not be easy'.
EU council chief Donald Tusk underlined the challenge she faces by immediately rejecting the idea. 'The backstop is part of the Withdrawal Agreement, and the Withdrawal Agreement is not open for renegotiation,' he said.
May repeated her invitation for Jeremy Corbyn to join talks on how to proceed with Brexit - and he grudgingly accepted.
She also played down the vote against no deal, saying: "I agree that we should not leave without a deal. However, simply rejecting no deal is not enough to stop it," she said.
The results came after May urged wavering Tories to hold off rebelling, promising it would not be their "last chance" to avoid no deal.
"I will never stop battling for Britain, but the odds of success become much longer if this House ties one hand behind my back," she said.
PM's final message
In a final message before the Commons showdown began, the PM admitted she must attempt to renegotiate the Withdrawal Agreement, and appealed for MPs to vote for an amendment that would give her a "mandate" to demand the Irish border backstop is replaced.
Earlier, May had struck a last-minute deal with Tory Brexiteers to rip up her agreement with the EU and return to Brussels to negotiate a new backstop.
Speaking in the Commons, the PM said she will attempt to renegotiate the Withdrawal Agreement and appealed for MPs to vote for an amendment that would give her a "mandate" to demand the Irish border backstop is replaced.
Opening the debate ahead of crucial votes tonight, May said: "We need to send a message about what we want."
But she also urged Tory Remainers to hold off from "tying her hands" by rebelling to support a separate proposal from Yvette Cooper that could delay Brexit, insisting this will not be their "last chance" to stop no deal.
Speaker John Bercow has set the stage for a titanic showdown by accepting the PM-backed amendment tabled by senior Tory Sir Graham Brady, as well as Cooper's plan backed by Jeremy Corbyn.
The climbdown on reopening her Brexit package came after Boris Johnson, Jacob Rees-Mogg and the DUP signalled they will only support the Brady amendment tonight if May made the pledge.
The premier had been dodging explicitly saying the package she had thrashed out with Brussels would be fundamentally changed - instead suggesting there could be "legally binding" add-ons.
May also vowed to look "seriously" at a Plan C hammered out by pro-EU and Eurosceptic Tories - which would involve demanding a much looser backstop, and if that could not be agreed asking for a longer transition period to seal other trade arrangements. Brexiteers believe approving the Brady plan could be a first step towards securing their vision.
But the votes tonight are on a knife edge, with Parliament seemingly no closer to breaking the Brexit deadlock.
Even if the Brady amendment is passed, May's leverage in negotiations with Brussels could be undermined if the Cooper plan is accepted by MPs.
The PM spoke on the phone to EU commission president Jean-Claude Juncker this morning, although the bloc is already dismissing the idea of reopening the Withdrawal Agreement.
May had been due to close the Brexit debate tonight with a final plea not to derail her strategy.
But instead she opened the session with a statement explaining why she is backing Sir Graham's amendment as a means to achieving her goals of overhauling the backstop.
She warned MPs that they could not simply keep saying things were unacceptable. "We need to send a message about what we do want," she said.
Allies hope that a strong vote by MPs will give her "firepower" to go back to Brussels and secure concessions that can satisfy the DUP and Brexiteers.
However, May is also facing a potentially catastrophic challenge from the Remainer wing of her party - which has been pushing her to rule out a no-deal Brexit.
The premier has been trying to reassure MPs that they will have another chance to stop the UK crashing out, and do not need to support the Cooper amendment, which would dramatically reduce her leverage in talks with the EU.
May said she will return to the Commons "as soon as possible" with a revised deal which will be subject to a "meaningful vote" by MPs.
If this is rejected by MPs, she will table a further amendable motion for debate the next day.
If no new deal has been reached with the EU by February 13, May will make a statement to the House that day and table an amendable motion for
Despite what she acknowledged was a "limited appetite" in Europe for reopening talks, she insisted: "I believe with a mandate from this House... I can secure such a change in advance of our departure from the EU."
The amendment from Sir Graham - the chair of the influential Tory backbench 1922 Committee - was one of seven selected by Mr Bercow for consideration and possible votes on Tuesday evening.
Also up for debate is a cross-party proposal from Labour MP Yvette Cooper and Tory Nick Boles - backed by the Labour frontbench - which would hand control over the Brexit process to Parliament, potentially delaying departure day from March 29 to the end of the year.
Bercow also selected Jeremy Corbyn's own amendment - which would allow debate on Labour's preferred plan or a second referendum - as well as others which would rule out a no-deal Brexit, extend the two-year negotiation under Article 50 or permit a series of 'indicative votes' to establish the will of the Commons.
In an attempt to fend off possible rebellion by Tories seeking to avoid a no-deal departure, May promised that Tuesday's votes would not be MPs' final chance to pass judgment on EU withdrawal.
She told the Commons she aims to return to the House 'as soon as possible' with a revised deal, which will be subject to a 'meaningful vote' of MPs. If defeated, she will table another amendable motion for debate the following day.
If no new deal is reached by February 13, the PM will make a statement to Parliament that day and table an amendable motion for debate the following day.
May acknowledged that the Commons defeat of the Withdrawal Agreement she agreed with EU leaders last November had been 'decisive'. And she told MPs: "I listened."
It was in the interests of the whole House to back the Brady amendment, which would resolve the main obstacle to Britain securing a smooth and orderly exit from the EU, she said.
May said it was time for Parliament to "speak as one".
"I will never stop battling for Britain, but the odds of success become much longer if this House ties one hand behind my back," she said.
"I call on this House to give me the mandate I need to deliver a deal this House can support.
"Do that and I can work to reopen the Withdrawal Agreement. Do that and I can fight for a backstop that honours our commitments to the people of Northern Ireland in a way this House can support. Do that and we can leave the EU with a deal that honours the result of the referendum.
"So the time has come for words to be matched by deeds. If you want to tell Brussels what this House will accept, you have to vote for it. If you want to leave with a deal, you have to vote for it. If you want Brexit, you have to vote for Brexit."
She warned those considering voting for the 'cacophony' of rival amendments to rule out no-deal: 'Unless we are to end up with no Brexit at all, the only way to avoid no-deal is to agree a deal.
"That is why I want to go back to Brussels with the clearest possible mandate to secure a deal that this House can support. That means sending the clearest possible message not about what this House does not want but what we do want."
