We are focused on getting any possible vote in favour of the PM's deal", he told BBC radio, adding, "I don't think the British public are served by fantasies about magical alternative deals.
But in a vote on Tuesday that raised the chances of a defeat in next week's historic vote, MPs including former ministers approved an amendment that will curtail the government's tax powers in case of a no-deal Brexit.
It came as the third day of debate was to due open on the draft deal the prime minister has struck with Brussels.
Mr Hunt warned that failure to deliver Brexit would be "incredibly damaging" for the United Kingdom and something the country would regret for "many, many generations".
In a speech on Thursday, Labour leader Jeremy Corbyn called for a general election "at the earliest opportunity" to "break the deadlock" over Brexit.
Hunt acknowledged that May's deal was "not perfect" but insisted it "broadly delivers Brexit".
Warning that there may be no consensus in the Commons around any possible outcome, the Foreign Secretary told Today: "If this deal is rejected, ultimately what we may end up with is not a different type of Brexit but Brexit paralysis".
Mr Hunt's words were seen as a way to win over hardline Brexiteers who are not convinced by Mrs May's deal.
The foreign secretary said that unless MPs delivered on the referendum result it would be a "breach of trust" with the electorate and bad for the country's reputation overseas.More news: China trade talks extended, 'going very well'
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"I believe the European Union will extend Article 50 for us but I think they would only do it in a very limited number of circumstances. and we need to explore what those circumstances may be".
He said: "If we were, as a political class, not to deliver Brexit, that would be a fundamental breach of trust between the people and the politicians".
"There has been a referendum in which people overall decided that we should leave the European Union and I have said repeatedly that I think the job of elected politicians is to look at the best way of extracting the best deal under those circumstances".
This latest vote would mark the government's second defeat in 24 hours, as May was humiliated on Tuesday when a powerful cross-party coalition of MPs voted in favor of an amendment to limit the government in preparing for a no-deal Brexit.
Mrs May was boosted on Thursday by two Tory backbenchers - her former policy adviser George Freeman, and Trudy Harrison - indicating they will back her deal, as well as by a call from Japanese PM Shinzo Abe for the United Kingdom to avoid no-deal.
Labour said it would try to trigger an election by calling for a no-confidence vote in the government if May's deal is defeated next week.
Ms Rudd said it was "right" for the Government to make preparations for a no-deal Brexit, comparing it to wearing a seatbelt when driving a fast auto.
London's Metropolitan Police said concerns about shortages of goods could lead "to a significant increase in customers", in a statement emailed to CNN and suggests stores should consider planning for extra security.
The Daily Mirror says Mrs May has "caved in" on workers' rights to save her deal.