MPs across the parties should try to “forge a consensus” over Brexit, the work and pensions secretary has said.
In the Daily Mail, Amber Rudd wrote: “Brexit is in danger of getting stuck.”
Foreign Secretary Jeremy Hunt said it would be possible to get “a version” of the prime minister’s Brexit deal approved by MPs.
But BBC deputy political editor John Pienaar says that without an end date to the controversial “backstop” plan, it has no chance of passing.
Theresa May’s bid to make her deal more acceptable to MPs suffered a blow when EU leaders said it was “not open for renegotiation”.
Last week, Mrs May delayed a vote in the House of Commons on her Brexit deal, fearing a heavy defeat.
She then went on to win a confidence vote brought by her own MPs – but vowed to listen to the concerns of the 37% of Tory MPs who voted against her.
She travelled to Brussels to make a special plea to EU leaders, to try to make her deal more appealing.
However, the EU said there could be clarification but not renegotiation.
By BBC political correspondent Tom Barton
A cabinet split? Certainly, a disagreement between senior ministers over the viability of Theresa May’s Brexit deal.
Jeremy Hunt, the foreign secretary, telling Radio 4’s Today programme that it will be possible to get “a version” of the prime minister’s Brexit deal approved by MPs.
Amber Rudd, the work and pensions secretary, openly musing in the Daily Mail that the government needs need “to try something different” because parliament is currently headed towards “no compromise, no agreement and no deal”.
For the foreign secretary, the risk of no deal works as leverage in the UK’s attempt to get the EU to reconsider the detail of the Northern Ireland backstop, warning European leaders that “they can’t be sure” Parliament would stop it.
For Amber Rudd, it’s something which “for the sake of all our futures, mustn’t be allowed to happen”.
Ms Rudd – who backed Remain in the referendum – said she supported Mrs May’s deal and advocated assembling a “coalition” to avoid what she called “the rocks of no deal”.
She said the country “will face serious trouble” if MPs “dig in against the prime minister’s deal”.
“We need to find a plan that a majority in Parliament can support,” she said.
“We need to try something different. Something that people do in the real world all the time, but which seems so alien in our political culture – to engage with others and be willing to forge a consensus.
“It also requires everyone to abandon outrage and accusations.”
Many of Mrs May’s own MPs are concerned that the controversial “backstop” plan – which is aimed at preventing a hard border in Northern Ireland – would keep the UK tied to EU rules indefinitely and limit its ability to strike trade deals.
Mr Hunt said the EU needed to listen to appeals from the British government to provide “legally enforceable language” that the backstop would be temporary.
He told BBC Radio 4’s Today programme: “The thing that the House of Commons will not accept is any risk of us being permanently trapped through the Northern Irish backstop in the customs union.”
He added: “The only way that we’re going to get through the House of Commons and give the British people the Brexit that they voted for is to have a version of the deal that the government has negotiated.”
However, the BBC’s John Pienaar said the problem remained that only “an end date or a key to the exit door” would make it possible for the deal to be supported by MPs.
He added: “The EU has shown no indication, publicly or privately at any point, that it is willing to give that.”
PM time-wasting ‘unacceptable’
Former Tory minister Jo Johnson, a Remainer who resigned over Mrs May’s handling of negotiations, said MPs should be able to vote on her Brexit deal next week.
He said it was “unacceptable” for the PM to “run down the clock” and leave Parliament with only a choice between her deal and no deal at all.
One idea, favoured by at least one cabinet minister, is a series of votes on other plans, such as a relationship similar to Norway’s with the EU, or another referendum, before next month’s “meaningful vote” in the Commons.
At a Leave Means Leave rally in London on Friday, former UKIP leader Nigel Farage told the BBC it was “outrageous” another referendum could happen, but added: “I can see where we’re going.”
Mr Farage added the treatment of Mrs May in Brussels this week had been a “shaming moment” for both the UK and the EU and that the prime minister’s Brexit deal was now “dead”.
Labour leader Jeremy Corbyn said the withdrawal deal was now “dead in the water”.
He added: “Rather than ploughing ahead and dangerously running down the clock, the prime minister needs to put her deal to a vote next week so Parliament can take back control.”