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The UK government insists Theresa May's Brexit proposal is a "workable, credible" deal, despite being rejected by EU leaders at a summit in Salzburg.
Minister James Brokenshire said "tough words" were to be expected near the end of negotiations but the government was "resolute" in its bid to get a deal.
He said the so-called Chequers plan "does deliver" and it was now for the EU to "be specific" about its concerns.
The UK is due to leave the EU on 29 March 2019.
Prime Minister Theresa May says her plan for the UK and EU to share a "common rulebook" for goods, but not services, is the only credible way to avoid a hard border between Northern Ireland and the Republic of Ireland.
But it is opposed by some within her own party who argue it would compromise the UK's sovereignty, and got a cool reception at this week's EU summit in Salzburg, Austria. Mrs May described her talks there with European Council President Donald Tusk as "frank".
In a press conference, Mr Tusk said that while there were some "positive elements" in the Chequers plan, EU leaders had agreed the proposals needed to be redrawn: "The suggested framework for economic cooperation will not work, not least because it is undermining the single market."
He followed it up by posting a photograph on Instagram of he and Mrs May looking at cakes with the caption: "A piece of cake, perhaps? Sorry, no cherries."
The EU has argued that the UK cannot "cherry-pick" elements from its rulebook.
On Friday, Communities Secretary Mr Brokenshire, a former Northern Ireland Secretary, told BBC Radio 4's Today the prime minister was taking a "firm resolute approach".
He said the UK believed that the Chequers deal did respond to the EU's concerns and that there was a desire to get a deal on both sides.
He added: "The Chequers deal is a workable, credible deal to meet our ambitions.
"They [the EU] have said that it's about the integrity of the single market and we believe the Chequers deal responds to that, and it's for the EU to engage with what's on the table."
But former cabinet minister Iain Duncan Smith, who campaigned for a Leave vote in the EU referendum, questioned why the UK had gone for a plan which "was so obviously not going to cut the mustard with the EU".
He added that the EU had "behaved appallingly" towards Mrs May and he described Mr Tusk's Instagram photograph as "quite insulting".
"I think they got personal," he said.
Both sides are trying to reach a deal in time and want to avoid a hard border - physical infrastructure like cameras or guard posts - between Northern Ireland and the Irish Republic but cannot agree on how.
The EU insists on its own "legally-binding Irish backstop" - what it describes as an insurance policy to prevent the return of physical infrastructure on the border, in the event no other solution can be found.
By BBC political editor Laura Kuenssberg
Theresa May and her government have been trying to pursue a middle way, to find a stance between the basic options - a close "Norway style", or free trade deal roughly like Canada.
It feels that the search for something else has been in vain.
Sources on the EU side express irritation at the UK's approach, at what they see as the strident tone the prime minister took in the last 48 hours.
The European Council is not the same as Prime Minister's Questions, it's suggested.
But to kick out publicly as they did in Salzburg certainly runs the risk of pushing Mrs May too far.
It suggests Northern Ireland should stay aligned with the EU in key areas, effectively staying in the customs union and single market and not needing those border checks. But the UK says this is unacceptable as it would split Northern Ireland from the rest of the UK.
Mrs May has said the UK will come forward "shortly" with new proposals on the so-called "backstop" arrangements, but also insisted Chequers was the "only serious and credible proposition" for an overall deal.
Transport Secretary Mr Grayling told BBC Newsnight: "At the moment what the European Union is asking in and around Northern Ireland is simply impossible for any UK government to accept. And actually if they stick with that position, there will be no deal," he said.
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Conservative MP Stephen Crabb told Today the EU's attempts to "belittle" Mrs May would push people who have been advocating compromise with the EU to think "the quicker we're out of this circus the better".
But Scotland's First Minister Nicola Sturgeon
: "Now that the EU has explicitly rejected it, the Chequers pretence has to stop. At the very least, single market/customs union membership must be back on the table and the Article 50 clock [the time-limited process taking Britain out of the EU] must be stopped to avoid a cliff edge."
Now that the EU has explicitly rejected it, the Chequers pretence has to stop. At the very least, single market/customs union membership must be back on the table and the Article 50 clock stopped to avoid a cliff edge.— Nicola Sturgeon (@NicolaSturgeon) September 21, 2018
Mr Tusk said on Thursday that October would be the "moment of truth" for reaching a deal, and that "if the conditions are there" an additional summit would be held in November to "formalise" it.
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