The last-ditch bid to make progress in talks came amid predictions the Prime Minister was headed for a second humiliating defeat on her Withdrawal Agreement and it remains to be seen if the concessions will be enough to see it pass through Parliament.
The Prime Minister was looking to downgrade her promised meaningful vote to a provisional vote on whether they would back attempts for technical changes to the deal that attorney general Geoffrey Cox has been struggling to negotiate with Brussels, reported a number media channels, including the Daily Telegraph, Guardian, Independent and Sky News, citing governments sources.
The campaign group, fronted by former Ukip leader Nigel Farage, has argued that not taking part in the elections could mean that the United Kingdom no longer has representatives in the European Union whilst Brexit negotiations are ongoing.
The House of Commons overwhelmingly rejected the deal in January, primarily because of concerns over arrangements for the Irish border. It honours the UK's solemn commitments in the Belfast/Good Friday Agreement.
"The PM's proposed deal is the worst deal and will keep us trapped as a satellite of the EU, paying at least £39 billion for the privilege and with no say in EU affairs, at the mercy of the EU for years to come".
Those changes would not affect the terms of the withdrawal, but offer legal assurances to back it up.
British foreign minister Jeremy Hunt warned on Sunday that Brexit is in peril if British Prime Minister Theresa May's Brexit deal is rejected.
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Senior Labour backbencher Yvette Cooper said the PM would be guilty of a "straight-up lie" if she failed to go through with votes allowing MPs to delay Brexit.
Because the majority of ERG Brexiter Tory MPs, perhaps 60 odd or more, would still vote against her Brexit plan.
They urged her to table a "conditional" motion setting out the terms for dealing with the backstop issue which parliament would be prepared to accept. "To stand here today and say this is a significant change when she's repeating what she said on 14 January is not going to take anyone here far".
Former chief whip Andrew Mitchell told the paper: "Anything that avoids what looks like a massive defeat on Tuesday is worth considering". Brexit Secretary Stephen Barclay accompanied the PM to the talks.
"Technical" talks aimed at securing concessions from the European Union failed to secure a breakthrough over the weekend.
But if Brexit is delayed and the United Kingdom is denied the right to elect new MEPs, the hardline Eurosceptic group says ministers would "deny democracy" to the British people.
Elections for new Euro-MPs must be held if MPs delay Brexit, pressure group Leave Means Leave has said.