By Guy Faulconbridge and Kate Holton

LONDON (Reuters) - Hours after a Brexit deal collapsed, British Prime Minister Theresa May came under pressure on Tuesday from opposition parties and even some allies to soften the EU divorce by keeping Britain in the single market and customs union after Brexit.

May's ministers said they were confident they would soon secure an exit deal, though opponents scolded the prime minister for a chaotic day in Brussels which saw a choreographed attempt to showcase the progress of Brexit talks collapse at the last minute.

The Northern Irish party that props up May's minority government said it was only shown the draft of a deal promising regulatory alignment for both parts of Ireland late on Monday morning.

In a sign of just how politically precarious May's Brexit balancing act has become, the Democratic Unionist Party (DUP) also said it had warned May that it would not support her legislation in parliament unless the draft was changed.

The opposition Labour Party said one way for alignment of Northern Ireland with the Republic of Ireland to become acceptable was for the whole of the United Kingdom to stay in the single market and the customs union.

"What an embarrassment - the last 24 hours have given a new meaning to the phrase 'coalition of chaos'," Labour's Brexit spokesman Keir Starmer told parliament. "Yesterday, the rubber hit the road: Fantasy met brutal reality."

"Will the Prime Minister now rethink her reckless red lines and put options such as a customs union and single market back on the table for negotiation?" Starmer asked.

Nicola Sturgeon, the leader of Scotland's devolved government, said May's failure could signal a push to keep Britain in both.

"This could be the moment for opposition and soft Brexit/remain Tories to force a different, less damaging approach - keep the UK in the single market and customs union," Sturgeon said on Twitter. "But it needs Labour to get its act together. How about it @jeremycorbyn?"

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Scottish Conservative leader Ruth Davidson, who has been tipped as a potential future leader of May's party, also suggested May should consider keeping the United Kingdom in the single market and customs union.

May has repeatedly said Britain will leave both groupings when the United Kingdom ends its membership of the EU at 2300 GMT on March 29, 2019. But she has also called for a bespoke economic partnership.

Brexit minister David Davis said voters had chosen to leave the EU and that included both the single market and the customs union.

Davis said the government would never allow one part of the United Kingdom to remain in the single market after Brexit, though he did allow that regulatory alignment for Northern Ireland could apply to the whole of the United Kingdom.

Sterling rebounded from a six-day low against the euro on Tuesday to trade flat on the day, with investors cautiously optimistic that a deal on opening up talks on post-Brexit trade would be reached by the end of the week.

DUP TAIL WAGGING THE DOG?

May, who is now scrambling to thrash out a deal with the EU while keeping Northern Ireland's DUP and her own party onside, may return to Brussels as early as Wednesday to continue talks, a Downing Street official said.

"We're very confident that we will be able to move this forward," finance minister Philip Hammond said as he arrived for a meeting with EU counterparts in Brussels.

A European Commission spokesman said it was ready to resume Brexit negotiations as soon as London signals it is ready.

But the EU will only move to trade talks if there is enough progress on three key issues: the money Britain must pay to the EU; rights for EU citizens in Britain and British citizens in the EU; and how to avoid a hard border with Ireland.

Britain must present the European Union with a good offer this week on the terms of its divorce from the bloc, or it will be too late for the EU to prepare by mid-December for the start of talks on a future trade deal, a senior EU diplomat said.

"The 'deadline of deadlines' is this week," the diplomat said.

All sides say they want to avoid a return to a hard border between EU member Ireland and the British province of Northern Ireland, which might upset the peace established after decades of violence.

But DUP leader Arlene Foster said she had told May on Monday that the party could not support her minority government's legislation unless the Irish border draft deal was changed.

"When we looked at the wording and saw the import of all that, we knew we couldn't sign up to anything that was in that text that would allow a border in the Irish Sea," Foster told RTE in an interview.

After May lost her party its majority in parliament in a June snap election, she is dependent on the DUP's 10 lawmakers in the 650-seat British parliament to ensure she can pass legislation.

Nigel Dodds, deputy leader of the DUP, said it would work for as long as needed to get the Brexit deal right but accused Dublin of acting in a reckless and dangerous way that was putting years of Anglo-Irish cooperation in danger.

Irish Prime Minister Leo Varadkar said London had to make the next move, while Labour's Starmer said the DUP tail was now "wagging the Tory dog".

"As things stand, the ball is very much in London's court. There is time to put this agreement back on track and we await to hear from London as soon as they're ready," Varadkar told parliament.

(Additional reporting by Elizabeth Piper, William James, Costas Pitas, Alistair Smout and Stephen Addison in London, Elisabeth O'Leary in Edinburgh and Padraic Halpin in Dublin; Editing by Angus MacSwan and Hugh Lawson)