Lewis said May told a meeting of her MPs that they should vote in favour of an amendment, proposed by senior Conservative lawmaker Graham Brady, which calls for the backstop to be replaced with "alternative arrangements".
She urged MPs to vote to rule out the backstop, which would see the United Kingdom in a customs union with the EU until a new trade deal is signed, "to be replaced with alternative arrangements to avoid a hard border [in Ireland]". The backstop would keep the U.K.in a customs union with the EU in order to remove the need for checks along the frontier between the U.K.'s Northern Ireland and EU member Ireland after Britain leaves the bloc. The Brexit deal between London and Brussels - overwhelmingly rejected this month by British MPs - contains a so-called backstop provision ensuring that if all else fails, the border will remain open.
Another amendment, rejecting a no-deal Brexit, also won the support of Parliament yesterday.
Some others seek to rule out a no-deal Brexit so Britain can't tumble out of the bloc on March 29 without an agreement in place to cushion the shock.
Despite celebrations in Brussels, Mrs May had a positive night in the Commons after MPs widely supported Graham Brady's amendment that hands the Government a mandate to renegotiate the Irish backstop.
She said such a change would not be easy and there was "limited appetite" on the part of the European Union, but she believed she could secure it.
But Barley said there was no room for substantial renegotiation of Britain's Withdrawal Agreement with the European Union and, while it could be flexible on Britain's departure date, London must have a plan for a delay to make sense.
Speaker John Bercow reportedly received 14 amendments to May's Brexit deal, with half a dozen of them expected to be put to a vote in order to achieve Brexit Plan B.
But an ardently pro-Brexit group of Conservative lawmakers led by backbencher Jacob Rees-Mogg said they would not back the move because it does not force the reopening of the withdrawal agreement.
Simon Bulmer, professor of European Politics at the University of Sheffield, says above all Brussels is seeking clarity from what it sees as disarray on the British side.
Lawmakers from different parties have submitted amendments that could be put to the vote on Tuesday and, if passed, would influence how the government proceeds with its Brexit plans.
Ireland's European Affairs Minister, Helen McEntee, said: "There can be no change to the backstop".
This proposal, tabled by Labour's Yvette Cooper and Tory Nick Boles, is one of the strongest-backed of a bevy of rival amendments tabled for debate.
Former Foreign Secretary Boris Johnson, who resigned previous year in opposition to the backstop and May's agreement, described the amendment's passage as "a victory for the prime minister, but also it's a victory for the U.K." as it allows May to return to Brussels stronger to negotiate changes.
May, who will close Tuesday's debate, now hopes her Tory party will say clearly what it wants to change in the deal she's struck with the EU.
At the weekend, Ireland's foreign minister, Simon Coveney, stated baldly the backstop simply "isn't going to change".
It comes ahead of crucial votes in Parliament on January 29.
One of the proposed amendments was presented by Conservative lawmaker Dominic Grieve who sought to give the parliament rather than the government control over the Commons' agenda on several days. What that basically means is that if there isn't anything else but the Brady Amendment being approved, we're still nowhere near getting a deal done before 29 March.