British Prime Minister Theresa May leaves Downing Street to face a vote of no confidence in the House of Commons on January 16, 2019 in London, England.

The BBC added that May held a conference call with the Cabinet, her closest ministers, on Sunday and said "it's understood she wants to show the EU that MPs could back a deal without a backstop, in the hope of encouraging Brussels to soften its position."

May is now focused on persuading Brexiteers within her Conservative party and the pro-Brexit Democratic Unionist Party (DUP) in Northern Ireland to back her deal by resolving Irish "backstop" concerns, according to the BBC on Monday, citing anonymous sources within the Cabinet, top-level ministers.

May has since held cross-party talks although opposition parties have called on her to both rule out a 'no-deal' Brexit (where Britain leaves the EU abruptly with no ongoing relationship in place) or to hold a second referendum . A second vote is the preferred option of several larger opposition parties, including the Liberal Democrats and Scottish National Party (SNP), both of whom oppose Brexit.

U.K. Prime Minister Theresa May is due to lay out a 'Plan B' for Brexit to lawmakers on Monday as she looks to secure a fresh withdrawal agreement before the March 29 deadline to leave the European Union (EU).

The Irish "backstop" – a fall-back option intended to keep the Irish/Northern Irish border open if the U.K. and EU fail to strike a trade deal in a 21-month transition period following the departure date in March 2019 – has been a key stumbling block in Brexit negotiations.

It's controversial as it would mean that Northern Ireland, a part of the U.K., stays roughly aligned to the rules of the EU's single market in a bid to avoid a "hard" border with Ireland. That has proved controversial for lawmakers across the spectrum and May was expected to focus on resolving the backstop issue in a bid to persuade politicians within her own Conservative party and the opposition to back her deal.

The U.K.'s international trade minister said on Sunday that the Brexit process was at risk of being "hijacked" by 'Remainers' within parliament. May is expected to address Parliament on Monday afternoon and to set out how she intends to deal with the Brexit deal impasse.

Remainers and Brexiteers remain at loggerheads over Brexit and what level of closeness the U.K. should have to the bloc in future, however.

"You've got a 'leave' population, and a 'remain' parliament," Liam Fox told the BBC on Sunday.

"Parliament has not got the right to hijack the Brexit process … What we are now getting are some of those who were always absolutely opposed to the result of the referendum trying to hijack Brexit and in effect steal the result from the people."

Meanwhile, the Shadow Brexit Secretary Kier Starmer said that the U.K. now needed to be "realistic about what the options are" when he spoke to the BBC's Andrew Marr show Sunday.

Kamal Sharma, director of G10 FX Strategy for Europe at Bank of America Merrill Lynch, told CNBC on Monday that the direction May's deal will take, and how the wider Parliament reacts to it, is crucial this week.

"There's been very interesting developments over the weekend, it does seem that the Conservative party still is grappling with itself rather than a negotiation between itself and the EU," he told CNBC's Squawk Box Europe.