Video report by ITV News Correspondent Angus Walker
A Canada-style Brexit trade deal that broke up the UK would be worse than no deal, Theresa May has said.
Brussels has shown hope, expectation and desire for an agreement, the Prime Minister insisted.
But she said any move that would put a border in the Irish Sea would be a “bad” outcome.
And she accused Labour of being ready to accept any proposals the EU put forward regardless of how harmful they were.
It comes after senior Tory Leavers including David Davis and Jacob Rees-Mogg backed proposals calling on the Government to seek a “basic” free trade agreement for goods of the kind struck between the EU and Canada and pointed out such a move had already been offered by Brussels.
Downing Street later said the proposals would mean “Northern Ireland effectively remaining in parts of the single market and customs union”.
Speaking to reporters as she flew to New York for the United Nations General Assembly, Mrs May said: “First of all, I have always said no deal is better than a bad deal.
“I think a bad deal would be, for example, a deal that broke up the United Kingdom.
“We want to maintain the unity of the United Kingdom.
“What we have put on the table is a good deal, it’s a deal which retains the union of the United Kingdom, our constitutional integrity, it’s a deal that provides for no hard border between Ireland and Northern Ireland, protects jobs and enables us to have a good trading relationship with Europe and also the rest of the world.
“When we get to the point of a deal – and as Prime Minister I do believe we can get to a good deal – we will take that deal back to Parliament and at that point MPs will have a clear choice.
“They will have a choice as to whether to support that deal.
“They will have to recognise looking at their vote that what we are doing is delivering on the vote of the British people in the referendum.
“What I do know is that we have a Labour Party whose position in this (is) clear.
“Labour will oppose any deal I bring back regardless how good it is for the UK.
“They will accept any deal the EU gives regardless of how bad it is for the UK.
“That is not in the national interest, what we are doing is in the national interest.”
European Council president Donald Tusk mocked Mrs May’s negotiating strategy on Twitter after last week’s Salzburg summit, posting a picture of a cake with no cherries in reference to claims the PM is trying to “cherry pick” the best aspects of the EU.
Asked if she wanted an apology, she replied: “What I welcome is the fact we have seen Donald Tusk following our response, we have seen Donald Tusk actually clarify the position that they were taking and make it clear that actually the EU side does want to look at a deal.”
Mrs May said if the EU had concerns about her Brexit Chequers plan for future relations it should set them out along with any counter-proposals.
She added: “What he has clarified is that there is hope and expectation and desire for a deal on the side of the European Union and there are aspects of the Chequers proposal which they are comfortable with.”
The Cabinet backed proposals for EU nationals to be subject to the same rules as migrants from the rest of the world after free movement ceases to apply in the UK post-Brexit.
The PM said: “I have been very clear that one of the things that’s non-negotiable is an end to free movement and that is absolute.
“People voted to bring an end to free movement and we will do that.
“As we look at our future proposals for our immigration rules, what we want to do is ensure that we are recognising the concerns that people have about immigration, that we are working to continue to bring net immigration down but obviously we do that in a way that’s going to be good for the British economy and our future.”