So far, May has shown little appetite to change tack in her strategy to leave the EU.
Critics of Theresa May's plans to leave the European Union stepped up the pressure on the prime minister on Sunday, with one former minister saying her cabinet team should exert its authority to force her to change course.
With less than six months before Brexit day and just days before May heads to Brussels for a summit, former Brexit minister David Davis pressed May to abandon her proposal for leaving the European Union, saying the bloc "has rejected it. The public does not like it. Parliament will not vote for it".
The head of a Northern Irish party that props up her government also raised the stakes, saying it was ready to trigger a so-called no deal Brexit to prevent what it called "the annexation" of the British province by the EU.
So far, May has shown little appetite to change tack in her strategy to leave the EU, pressing her plan and trying to persuade lawmakers in her Conservative Party and in opposition Labour to vote for any deal based on it in parliament.
Davis, who resigned his post in July, criticised the government for accepting "the EU's language on dealing with the Northern Ireland border" last December, said it was now up to senior ministers to use their influence.
"This is one of the most fundamental decisions that government has taken in modern times. It is time for cabinet members to exert their collective authority," wrote Davis, who campaigned for Britain to leave the EU in the 2016 referendum.
"This week the authority of our constitution is on the line," he said in an article in the Sunday Times.
Lobbying of May from all sides of the Brexit debate has increased in recent weeks as London and Brussels edge closer to an agreement on a draft withdrawal treaty to cover the divorce terms, a transition period and a solution for Northern Ireland.
Preventing any return of a hard border between the British province and EU member Ireland has become one of the major obstacles to such an agreement, with Brexit campaigners fearful that a non time-limited backstop will keep Britain inside a customs union with the bloc indefinitely.
May insists any customs arrangement as part of the backstop must be temporary, but the EU has refused to set an end date.
Health minister Matt Hancock suggested the backstop could be temporary without such a date, an argument that may fall flat for some eurosceptic lawmakers who are calling for May to "chuck Chequers", her Brexit plan named after her country residence.
"There are different ways to ensure that something is time limited. There are different ways of doing that. For instance, you can set conditions at the point at which the arrangements come to an end," he told the BBC.
Even if May reaches agreement with the Brussels on a withdrawal agreement, she will face a struggle to get any deal through parliament and may even find opposition from her Northern Irish partners to other legislation such as the budget.
"I fully appreciate the risks of a 'no deal' but the dangers of a bad deal are worse," Arlene Foster, head of the Democratic Unionist Party, wrote in an article in the Belfast Telegraph.
"This backstop arrangement would not be temporary. It would be the permanent annexation of Northern Ireland away from the rest of the United Kingdom and forever leave us subject to rules made in a place where we have no say."
Eurosceptic lawmakers in her Conservative Party are also concerned about a backstop, fearing it could keep Britain in the EU's sphere of influence possible forever and stop the country from striking new trade deals with the rest of the world.
Davis' words were clearly targeted at several Brexit supporters in May's cabinet of top ministers after there were reports that some were considering their position after being updated on the state of the Brexit talks.
But Iain Duncan Smith, a former Conservative leader and eurosceptic lawmaker, urged caution, saying it was not time to try to bring down the prime minister and replace her with someone else.