The backstop is an insurance policy aimed at avoiding controls on the sensitive border between the British province of Northern Ireland and European Union member Ireland, though some British lawmakers worry it could trap the United Kingdom in the EU's orbit indefinitely.
May has also spoken to European Commission President Jean-Claude Juncker amid speculation the British leader might dash to meet Juncker and other EU leaders in France.
"We're very clear that the Withdrawal Agreement can't change but we want to try to be helpful in terms of providing the clarity and reassurance that's needed in Westminster that the backstop is meant to be temporary", Ireland's Coveney said. "It says that we will work together, in good faith, in pursuit of a future relationship that ensures that the objectives of the protocol, particularly the need to avoid a hard border, are met".
Hard-core Brexit supporters in May's Conservative Party and the prime minister's allies in Northern Ireland's Democratic Unionist Party both said they could not support the deal, which Parliament rejected by an overwhelming margin in January.
If "no deal" is off the table, 14 March will see British lawmakers vote for the delay of Brexit.
"We have secured legal changes", May said in a late night news conference in Strasbourg beside Juncker, 17 days before the United Kingdom is due to leave the European Union on March 29.
"Having studied the documents, I would be surprised if they are sufficient to enable the Attorney General to change the central plank of his December legal advice", Starmer said on Twitter.
The Commons motion the PM wants MPs to back describes the first new document as "the legally binding joint instrument" relating to the Withdrawal Agreement.
The Times urged MPs to consider the deal against a wider backdrop of "global instability" - and said it is "farcical" that they will vote on a package "they will have had just hours to assess". This is what Brexit hardliners are hoping for - that they will get another chance to force May to take the United Kingdom out of the bloc with no deal.
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Former Labour Cabinet minister and chair of the influential Commons Home Affairs Committee Yvette Cooper, who has spearheaded parliamentary efforts to rule out a no-deal exit, called for talks on the withdrawal deal and the UK's exit to be put on hold while May tries to build a consensus in Parliament and the country.
According to the Wall Street Journal today reporting the cost of Brexit turmoil: "There is one certainty in Brexit: London's pre-eminent role in global finance has been diminished".
If May's Brexit deal gets rejected again, another vote will likely take place on Wednesday on whether the United Kingdom should leave the bloc without a deal.
"I have got to say that if you look at what the prime minister has said so far it seems to fall short of what she, herself, had promised".
Britain's political crisis sparked anxiety across the European Union on Wednesday as fears rose that Britain would crash out of the bloc on March 29 without a withdrawal agreement to smooth the way.
May offered lawmakers a "meaningful" vote on what she had hoped would be a revised deal on Tuesday but with no major changes yet secured, Brexit-supporting lawmakers warned it would be defeated again. It is the main sticking point for the majority of British lawmakers.
Some British lawmakers underscored that warning, telling their Brexit-backing colleagues that rejecting the deal could lead to Britain's departure being postponed indefinitely.
As the U.K. will leave the bloc at the end of March, the border between Northern Ireland - a U.K. territory - and the Republic of Ireland, will remain as the only land border between the U.K. and the EU.
"What then happens with local people wanting to go about their business, wanting to get to hospitals, wanting to get their kids to school, all of that kind of stuff?" he said.