The Prime minister is considering her next move after her European Union withdrawal plan was rejected by MPs for a third time.
Mr Gauke said Theresa May would have to look “very closely” if MPs back a customs union, thought to be the most likely preference, in a fresh round of indicative votes on Monday.
“I think that she would need to look very closely at that,” he told BBC1’s The Andrew Marr Show.
He warned it would “not be sustainable” to ignore MPs if they voted for a softer Brexit.
“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.”
Mr Gauke said he could not remain a member of the Government if it tried to leave with no deal, but said that Mrs May had made clear that was something she would not do.
“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 the Government that pursued that as a policy. That is a point I have made on a number of occasions,” he said.
“The Prime Minister has been very clear that when Parliament is making it clear what it wants to do, she is not going to go down that route.”
He said he did not favour a customs union as it went against the Tory party’s general election manifesto.
He added the prime minister’s deal was “the best outcome”.
“Sometimes you do have to accept your second or third choice to avoid an outcome you consider to be even worse,” Mrr Gauke said.
His comments follow reports that Mrs May’s aides are at loggerheads this weekend over whether to accept a soft Brexit involving a customs union or hold a snap poll if her deal is rejected by MPs for a fourth time.
The prime minister has until 12 April to seek a longer extension to the Article 50 process if the UK is to avoid leaving the EU without a deal.
The current deal is opposed by parties including Northern Ireland’s DUP and a group of her own MPs.