The Prime Minister has been asked to throw her support behind the changes, which could see the backstop limited by 12 months or banished completely. According to The Sun, some Brexiteers in the Conservative party are planning to table the amendment to the withdrawal agreement. They see it as a chance to remove the insurance plan that sees Britain tied to the EU.
Sources say if the deal is approved by Parliament a major row could escalate with the EU.
Sir Brady said: “My amendment on January 29 demonstrates that a majority can be found for the Withdrawal Agreement, as long as the backstop cannot be a permanent state.
“It is difficult to see how introducing a bill in exactly the same shape as the current Withdrawal Agreement could secure a majority, as it has already been rejected three times.
“So I have urged the Government to build on the success of my amendment by incorporating the necessary reassurance about the backstop on the face of the bill.”
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Theresa May’s Brexit deal has been voted down three times by Parliament so far, with many seeing it as a bad deal that leaves the UK shackled to the EU.
Former Tory minister Sir Mike Penning said it was a “golden opportunity to lance this boil” and stop the Brexit rot.
He added: “The EU have spent months asking us for a majority for a Brexit deal. Losing the backstop is the only way we will achieve one.
“It transforms the equation into Europe blocking a deal, rather than the British Parliament.”
The comments come after Mrs May was reported to give MPs another vote on her deal as early as next week.
She will once again use the promise of her quitting even sooner than planned in a bid to get her deal passed by the House of Commons.
ITV political editor Robert Peston said: “I am told the Prime Minister is giving serious consideration to introducing the Withdrawal Agreement Bill to the Commons but with the Northern Ireland backstop stripped out – in the hope it shows, like the Brady amendment, that her Brexit would be approved without it.
“We’re still chasing unicorns, and eating up time’ said a minister.”