Even a non-binding vote of no-confidence from a majority in the House of Commons could leave May's authority further weakened - and potentially prompt another vote in her entire government and a general election.
UK Prime Minister has until now survived one no-confidence motion against her while another motion was tabled by the Leader of the Opposition, Jeremy Corbyn, in the Commons on December 17.
Labour has resisted the calls, saying they won't try and collapse the Government until they know they can win and after MPs have had the chance to vote on May's Brexit deal.
An extraordinary public row broke out Sunday between British Prime Minister Theresa May and former Labour party leader Tony Blair over her Brexit deal.
Despite the impending deadline, it is still unclear whether the United Kingdom will leave under the terms of a deal, with no deal, or even whether it will leave at all, after the European Court of Justice said Article 50 could be revoked unilaterally and continue as a full EU member without seeking approval from Brussels.
Adding that Britain will have a positive future post-Brexit if it gets a deal, she said "we have to put politics aside" and talk across the House to build a consensus, putting "country before politics".
A growing number of politicians have been calling for a new referendum as May is struggling to get MPs on board with her Brexit deal and European Union leaders are refusing to renegotiate the deal.
The British Chambers of Commerce forecast on Tuesday that economic growth this year and in 2019 looks set to be the weakest since Britain emerged from recession in 2009, due to a freeze in business investment and weak consumer demand ahead of Brexit.
"Another vote which would do irreparable damage to the integrity of our politics, because it would say to millions who trusted in democracy, that our democracy does not deliver".
"Another vote which would likely leave us no further forward than the last".
He added: "Jeremy has taken the sense of the House of Commons which is clearly demanding a vote this week".
100 days are left for Britain to leave the EU.
"Cabinet agreed that, with just over three months until our exit from the European Union, we have now reached the point where we need to ramp up these [no-deal] preparations".
Mrs May's official spokesman told a Westminster media briefing that there were "no plans" to stage an indicative vote on a range of Brexit options, but did not definitively rule the option out.
Ms Rudd's intervention came after the Prime Minister's plans for securing reassurances from Brussels on her Brexit deal were mocked as "a wee bit nebulous" by Nicola Sturgeon.
"That's why it is more important than ever that the devolved administrations get behind this deal and listen to businesses and industry bodies across all four nations who have been clear that it provides the certainty they need".
Downing Street said that advice on no-deal preparations will be going out to households by various channels over the coming weeks.
The PM's key aides - her chief of staff Gavin Barwell and de facto deputy PM David Lidington - were accused of exploring the possibility of a "People's Vote" to break the Brexit deadlock, The Sun reports.
He had previously said that despite seeing off the motion of no confidence by 200 votes to 117 she should resign "as soon as the Queen has a moment in her diary to see her".
I think we played the first time against each other when I was eight and she was nine, something like that. Monday's Australian Open first-round match against Spaniard Roberto Bautista Agut could be his last.
If you think Part 1 is enough, you are wrong because we will never get enough of these Pacquiao vs Broner predictions. Jarrett Hurd - It will be a great fight but I want to go with my guy Broner because he has the skills to do it.
With the impressive win, San Antonio improved to 25-18 on the season and 18-5 at home where they've now won seven in a row. He did it last year, and he's doing it this year, too. "He's been fantastic, and obviously he had a great night".
When you're making adjustments when the defense is out there. "I have had the opportunity to be here a few other times. LaFleur will now get to work putting together his coaching staff as the busy offseason continues.
Ivanka Trump could be the next head of the World Bank along with other nominees including Former UN Ambassador Nikki Haley . The department received "a significant number of recommendations", a spokesman said.
For the president, who visited the Texas border with Mexico on Thursday, the border situation amounts to an invasion by criminals. Constitution to control government spending - and make it easier for a future Democratic president to bypass Capitol Hill.
Each player's match was attended by a member of the Armenian ring, who made sure the player complied with the agreement. Authorities said the players tainted results after taking bribes from an Armenian ring that was dismantled in October.
Speaking of New Orleans, the Saints are 5-0 in the Superdome during the playoffs with Sean Payton as their head coach. The magic of Nick Foles will continue for another week as I stand by what I said in my previous week of predictions.
Chelsea forward Alvaro Morata could be make stunning return to La Liga to join Spanish champions Barcelona , according to reports. In addition to Munir , Sevilla also announced Friday an agreement had been reached for Ajax centre-back Maximilian Wöber .
PA slams Israeli "apartheid road" in West Bank
The Palestinian Authority said in a statement that the road posed a challenge to "the credibility of the worldwide community". Segregated roads are common in the West bank, but until now, none are divided along their entire length by a wall.
What did Prince William do on Duchess Kate's birthday?
The Duchess of Cambridge celebrated her 37th birthday with a family tea party at Kensington Palace , it has been reported. But since becoming a mother, the Duchess has preferred to celebrate privately with her close family.
Bungie and Activision are Splitting Up
It's not hard to see why investors would sell their shares in ATVI, Destiny is a hugely popular video game franchise. As Bungie takes over complete control of Destiny , Activision will increase it's focus on its other IPs.