A no-deal Brexit would be better for the UK than any Canada-style free trade agreement allowed by the EU, Theresa May has argued as she arrived at a United Nations summit determined to regain control of the narrative after some bruising recent days for her own Chequers proposal.

At the start of a two-day trip to New York, the prime minister knocked down the possibility of a more basic free trade deal with the EU touted by more hardline Brexiters in her Conservative party, saying it could prompt the disintegration of the UK.

May – who criticised Labour’s Brexit plans as “not in the national interest” – also robustly defended her aim for continued alignment with EU standards in areas such as livestock and agriculture, saying the UK would want to maintain these anyway, and that they should not be a barrier to outside trading plans.

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“If having those EU regulations stopped good trade deals, how come the EU has done trade deals with countries around the rest of the world?” May said, a comment likely to raise some eyebrows among some strongly pro-Brexit Tory MPs.

On Monday a series of hardline Tories, including the former Brexit secretary David Davis, and Jacob Rees-Mogg, endorsed a Canada-style free trade alternative to Brexit, arguing it was the only way the UK could reap the full benefits of departure.

Speaking to reporters on her flight to the UN’s general assembly, where she will meet Donald Trump and the Iranian and Turkish presidents among other assembled leaders, May said the Brexiters’ plan would necessitate a hard Irish border and thus invoke the EU’s so-called backstop, which would keep Northern Ireland within elements of the customs union and single market, effectively drawing a customs border in the Irish Sea.

A no-deal departure would be preferable, she said: “First of all, I’ve always said no deal is better than a bad deal. I think a bad deal will be a deal, for example, that broke up the United Kingdom. We want to maintain the unity of the United Kingdom.

“What we’ve put on the table is a good deal. It’s a deal that retains the unity of the United Kingdom, our constitutional integrity. It’s a deal which provides for no hard border between Northern Ireland and Ireland, protects jobs and enables us to have a good trade relationship with Europe and good trade relationships with the rest of the world.”

At an informal EU summit in Salzburg last week, May had hoped to receive supportive words for her Chequers plan before what is expected to be a bruising Conservative party conference starting on Sunday. Instead, the prime minister was told that Brussels still had significant and fundamental problems with the plan.

Asked whether she continued to face the impasse outlined in her robust statement at Downing Street the day after the summit, May said subsequent words from Donald Tusk, the president of the European Council, had showed signs of progress.

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“I think what he has clarified is that there is hope and expectation for a deal on the side of the European Union, and of course there are aspects of the Chequers proposal which they’re comfortable with,” she said.

There was, however, more to be done, May said: “I believe it would work not just for us but for the EU. But if they have concerns they need to detail those concerns to us, and if they have counter-proposals let’s hear the counter-proposals, and then we can discuss those and take it forward.”

Asked about Labour’s ideas for Brexit that emerged from its conference this week, May condemned the expectation that the party’s MPs will vote against her final deal.

“From the Labour party’s point of view they will oppose any deal I bring back, regardless of how good it is for the UK, and they will accept any deal Europe gives, regardless of how bad it is for the UK,” she said. “That’s not in the national interest. What we’re doing is in the national interest.”

May, who arrived in New York on Tuesday lunchtime local time, was scheduled to meet the presidents of Iran, Turkey and France – Hassan Rouhani, Recep Tayyip Erdoğan and Emmanuel Macron – and the Canadian prime minister, Justin Trudeau.

On Wednesday May will meet Trump, address the general assembly, and discuss Brexit and trade with business leaders.