She said that, following this month's crushing 230-vote defeat in the Commons, it was clear that only reopening the agreement and changing the backstop would win MPs' support for her European Union withdrawal deal.
Labour leader Jeremy Corbyn, who has refused to join talks with May to find a way out of the impasse, asked which of the red lines she had changed since the deal was defeated two weeks ago: "She is refusing to accept the clearly stated will of this House". "Today we need to send an emphatic message about what we do want".
Or. perhaps May wasn't fibbing when she said she believed she could reopen negotiations in Brussels and get the European Union to agree to dump the Irish backstop.
Brady said if the motion was approved by Parliament, it would give Mrs May "enormous firepower" to go back to Brussels and renegotiate the Brexit divorce deal.
"She said a vote of the Brady amendment makes it clear that the current nature of the backstop is the key reason that the House can not support the deal", added the spokesman.
Mrs May is due to open this afternoon's Brexit debate in the Commons with a statement explaining why she is backing an amendment tabled by Tory MP Sir Graham, which would add a time-limited deadline on to the backstop arrangement.
However, the vote is not legally binding - meaning it showed the view of the House of Commons but does nothing to change the Brexit date of March 29.
But if that proves impossible, Britain would remain aligned with the EU customs union after 2020, while Northern Ireland would further stay aligned to some single market rules, in order to keep the Irish border open.
Yvette Cooper, a prominent lawmaker from the main opposition Labour Party, has placed an amendment that attempts to reduce the chance that Britain could leave the European Union without an agreement on future relations by compelling Prime Minister Theresa May to ask the bloc for more time if she fails to secure a deal by February 26.
These call for the backstop to be renegotiated or, if that fails, for Britain to leave on World Trade Organization rules at the end of 2021.
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The Labour leader said he would meet May, having refused to do so for a fortnight, because "parliament has voted to remove the immediate threat of crashing out without a deal on 29 March".
"The backstop is not a prerequisite for the future relationship", she said.
Members of Parliament have put forward more than a dozen amendments ahead of a debate on the legislation Tuesday, and house Speaker John Bercow is expected to allow several to be considered.
It was another win for the Government on the fifth amendment with MPs voting against Labour MP Rachel Reeves' plan by 322 to 290, majority 32.
The prime minister's spokesman said Mrs May would "engage" with her colleagues to discuss the plan, but she would also be looking at a possible time limit or a unilateral exit clause to the backstop as other "alternatives".
The DUP says it will back a Brexit compromise both Conservative remainers and leavers support.
Hancock said the pharmaceutical industry was doing everything it could to prevent shortages of medicines resulting from any delays at borders resulting from a no-deal Brexit.
She told her Cabinet in the weekly meeting she is ready to return to Brussels and seek legally-binding changes to the backstop.
On the eve of parliament's vote on unlocking the Brexit impasse, 10 food chiefs plus industry body the British Retail Consortium (BRC) called on MPs to work "urgently to find a solution that avoids the shock of a no-deal Brexit on March 29 and removes. risks for United Kingdom consumers".