Theresa May has said “very good progress” has been made on Brexit since the ill-tempered Salzburg summit last month, as she arrived in Brussels for talks with EU leaders and urged both sides to work “intensively” in the coming days.

The prime minister shrugged off the setbacks of last weekend and gave a defiantly upbeat assessment of the outlook for a deal after a brief meeting with the EU commission president, Jean-Claude Juncker.

“I believe a deal is achievable and now is the time to make it happen,” she said. “What we’ve seen is that we’ve solved most of those issues in the withdrawal agreement. There is still the issue of the Northern Irish backstop, but I believe everybody round the table wants to get a deal, and by working closely and intensively we can achieve a deal.”

May will be given half an hour on Wednesday evening to make Britain’s case before discussions continue over dinner without her.

EU leaders are concerned about the political constraints May faces at home, after ministers reiterated their concerns about the backstop at cabinet on Tuesday.

Senior EU diplomats said that given the impasse in the talks there would be a “major call” by a number of leaders over dinner for a no-deal summit to be pencilled into the calendar.

Lithuania’s president, Dalia Grybauskaitė, said it was difficult for the EU to negotiate “with a person that does not have full support”, in what appeared to be a reference to May. “If you have a negotiator on the other side of the table that has no strong mandate, it is very difficult for us to negotiate,” she said.

“Today, we do not know what they want. They do not know themselves what they really want. That is the problem.” she added.

Peter Pellegrini, Slovakia’s prime minister, said: “I am an optimistic person but not so today … We should be prepared for a no-deal result and maybe it will finish like that. My hope was that today we would have a conclusive solution on the table but it looks like it will not be a deal today.”

Xavier Bettel, the prime minister of Luxembourg, suggested conditions were not right for a summit next month to finalise a deal, as had been the original intention. “I am able to come back as often as we need to come to find a solution but I am not here to come to have a cup of coffee and shortbread,” he said. “We all have to move together.”

May declined to comment when asked about the possibility that the UK would seek an extension of the 21-month transition period. The EU has suggested extra time to seal a comprehensive trade deal could offer some further reassurance that the so-called backstop solution for avoiding a hard border on the island of Ireland would not be used.

Mark Rutte, the Dutch prime minister, said he was “cautiously optimistic” that an agreement would be reached “in the coming weeks” on the Irish border.

Quick guide Brexit and backstops: an explainer Show Hide A backstop is required to ensure there is no hard border in Ireland if a comprehensive free trade deal cannot be signed before the end of 2020. Theresa May has proposed to the EU that the whole of the UK would remain in the customs union after Brexit, but Brussels has said it needs more time to evaluate the proposal. As a result, the EU insists on having its own backstop - the backstop to the backstop - which would mean Northern Ireland would remain in the single market and customs union in the absence of a free trade deal, prompting fierce objections from Conservative hard Brexiters and the DUP, which props up her government. That prompted May to propose a country-wide alternative in which the whole of the UK would remain in parts of the customs union after Brexit. “The EU still requires a ‘backstop to the backstop’ – effectively an insurance policy for the insurance policy. And they want this to be the Northern Ireland-only solution that they had previously proposed,” May told MPs. Raising the stakes, the prime minister said the EU’s insistence amounted to a threat to the constitution of the UK: “We have been clear that we cannot agree to anything that threatens the integrity of our United Kingdom,” she added.

Senior diplomats said extending the Brexit transition would leave the UK paying billions more to the EU while having no say over how the money would be spent.

“We can assume there will be no rebate, for example,” said a senior EU diplomat, referring to the discount on budget payments secured by Margaret Thatcher in 1984.

The British rebate was worth £5.6bn in 2017, while the UK’s net contribution to the EU budget (excluding money for British farmers and regions) was £8.9bn.

Michel Barnier, the EU’s chief negotiator, told EU ministers on Tuesday that both sides had discussed extending the transition by one year to 2021.

Before the weekend it had been hoped that a text of the Northern Ireland backstop could be agreed at this week’s summit and a draft of the political declaration setting out the future relationship between the two sides could be tabled for discussion.

But after UK ministers expressed their reservations about potential compromises over the backstop, the Brexit secretary, Dominic Raab, dashed to Brussels on Sunday and it became clear there was not yet a version of the backstop on which both sides could agree.