With MPs set to hold a second round of “indicative” votes on alternatives to the Prime Minister’s deal on Monday, Justice Secretary David Gauke said she would have to “look closely” at any option that could command a majority.
After 170 Tory MPs - including 10 members of the Cabinet - wrote to Mrs May urging her to take the UK out of the EU quickly as possible, Mr Gauke reiterated he would resign rather than support a no-deal break.
Despite MPs rejecting Mrs May’s deal for a third time on Friday, Downing Street has made clear that she intends to to bring it back to the Commons for a fourth vote - possibly on Tuesday or Wednesday.
Mr Gauke said that while he believed it still remained the best option, the Government had to accept that - so far - it did not have the numbers to get it through Parliament.
He said that if MPs did coalesce around a plan by veteran Tory Ken Clarke for a customs union - which came closest to securing a majority in the last round of votes - ministers should be prepared to consider it.
Mrs May has so far strongly rejected the idea of a customs union, saying it went against the Conservative general election manifesto and would prevent Britain striking trade deals around the world.
Any move to accept a customs union would infuriate Brexiteers and would almost certainly lead to ministerial resignations.
However Mr Gauke told BBC1’s The Andrew Marr Show: “If Parliament is voting overwhelmingly against leaving the European Union without a deal but is voting in favour of a softer Brexit, then I don’t think it’s sustainable to ignore Parliament’s position and therefore leave without a deal.”
He added: “I think we also have to recognise my party does not have the votes to get its manifesto position through the House of Commons at the moment.
“We are in an environment when it is not just about going for your first choice. Sometimes you do have to accept your second or third choice in order to avoid an outcome you consider to be even worse.”
His comments come amid fears among Brexiteers that the indicative vote process - controlled by a cross-party group of MPs led by former minister Sir Oliver Letwin - will inevitably lead to a softer Brexit.
No 10 hopes that pressure will lead to more Brexiteers reluctantly backing Mrs May’s deal after it cut the majority against it from 230 and 149 in the first two votes to 58 on Friday.
Mr Gauke, who along with fellow pro-EU ministers Amber Rudd and Greg Clark, previously made clear he could not support a no-deal Brexit, said he would resign if that happened.
“My position is that it is not the responsible thing for a government to do, to leave without a deal in these circumstances, so obviously I wouldn’t be able to remain a member of the Government that pursued that as a policy,” he said.
With the Government now up against a new EU deadline of April 12, ministers played down suggestions that Mrs May could call a snap general election to break the deadlock.
However, former prime minister Sir John Major warned that if Parliament was unable to agree a way forward there could come a point where there would have to be a government of national unity.
“I think it would be in the national interest to have a cross-party government so that we can take decisions without the chaos that we’re seeing in Parliament at the moment where every possible alternative is rejected,” he told The Andrew Marr Show.
Mr Gauke said he did not believe the idea was workable, saying: “I don’t in all honesty think that it is practical.”
Meanwhile, shadow foreign secretary Emily Thornberry launched a highly personal attack on Mrs May, saying her judgment was impaired and she was “out of control”.
“Even with just days to go she is just saying ‘It is my deal or no deal’. That is not meaningful, that is not democracy. That is Theresa May stamping her feet and saying ‘I want this, no one else is allowed to do anything,’” she told Sky News’s Sophy Ridge On Sunday.
“No wonder she is in trouble. She is out of control. She is not listening to anyone. No one knows what it is that she is going to do next. I think her judgment has been undermined.”