Hopes are fading that a cross-party Brexit deal could be struck within days after Labour sources described Tuesday’s three-hour round of talks as “the most robust to date”.

Labour’s negotiating team was irked by leaks to Sunday newspapers about concessions Theresa May was preparing to make, which they believe amounted to little more than a rehash of the existing deal.

According to Labour sources, John McDonnell and Keir Starmer spent much of the meeting unpicking the government’s claim to have softened its position in recent weeks, with the atmosphere becoming “quite tense” at times.

By contrast, a Downing Street spokesman described the discussions as “constructive and detailed”, adding: “The teams have agreed to meet again for follow-up talks tomorrow afternoon, recognising the need to resolve the current Brexit deadlock in parliament,”.

Before Tuesday’s round of talks took place, May formally conceded it was now impossible to reach agreement in time to avoid participating in the European elections due this month.

With talks set to resume on Wednesday, both sides say they expect to decide within the coming days whether the “landing zone” for a deal is within reach.

The shadow business secretary, Rebecca Long-Bailey, said: “As we’ve said previously, the government needs to move on its red lines. We expect to make compromises but without a government that is willing to compromise it’s difficult to see how any agreement can be reached and I think the government is aware of that.”

May and Jeremy Corbyn are both constrained in how far they can go by tensions in their respective parties. May’s backbenchers have urged her not to concede on a customs union, and many Labour MPs say they will not back a deal without a referendum attached.

At the weekly cabinet meeting on Tuesday, the Brexit secretary, Steve Barclay, mooted the idea of ramping up no-deal Brexit planning. If no agreement can be reached and no majority found in parliament by 31 October, the government could have to request a third extension from the EU27.

There is growing anxiety that exasperated EU governments may decide to let the UK go. Barclay is expected to present a paper on no-deal planning next week.

The government is keen to introduce the withdrawal act implementation bill as soon as possible so that it can show it is pressing ahead with Brexit before the European elections.

Ministers had hoped Labour would agree to back the bill in its early stages in exchange for a promise of later amendments on issues including workers’ rights, but Labour negotiators have rejected that approach.

Government sources have suggested that if the talks fail to show clear progress soon, they could move on to the prime minister’s plan B: holding a series of votes in parliament to find a deal that could command a majority.

Starmer, the shadow Brexit secretary, said as he entered the talks that it was “crunch time”. After weeks of holding out the prospect of reaching a deal in time to avoid the UK taking part in the EU-wide elections, the government made a formal statement on Tuesday admitting that was now impossible.

“The PM deeply regrets that we haven’t left the EU on time, which was by 29 March, and that a consequence of that is that we will have to take part in the European elections,” May’s spokesman said. “She understands that many members of the public will feel great frustration at this.”

Q&A How do European parliamentary elections work in the UK? Show The UK elects 73 members (MEPs) to the European parliament, which is made up of 751 MEPs elected by the 28 member states of the EU. The UK is split into 12 European electoral regions, and each region is represented by between three and 10 MEPs. The constituencies are:

South East England (10 MEPs)

London (8)

North West England (8)

East of England (7)

West Midlands (7)

South West England (including Gibraltar) (6)

Yorkshire and the Humber (6)

Scotland (6)

East Midlands (5)

Wales (4)

North East England (3)

Northern Ireland (3) You can find out who is standing for election in your area here. In England, Scotland and Wales, voters can choose to vote for one party or individual. The ballot paper lists the parties standing with the names of their potential MEPs, as well as any individuals who are standing as independent candidates. The D’Hondt method of proportional representation is used to calculate how many seats each party or individual receives. In Northern Ireland, the single transferable vote method is used, where each voter ranks candidates in order of preference, marking 1 beside their most preferred candidate, 2 beside their second choice, and so on. These votes are then used to allocate Northern Ireland’s three MEPs. Those elected as MEPs on 23 May will represent the UK when the new European parliament assembles on 1 July, until such time as the UK ceases to be a member of the European Union.

He said parliament had had several opportunities to vote to leave the EU, “but sadly none of those votes have passed. We now wish to see the withdrawal agreement introduced, and ratified by parliament, as soon as possible – and ideally before MEPs take their seats.”

It is understood that the Conservative chairman, Brandon Lewis, is still deciding whether the party should hold a European elections launch, with activists and MPs reluctant to hit the doorsteps with no clear policy on Brexit.

Senior Tory officials are debating how big a role May should play in the contest and how far she should go in attacking Nigel Farage’s Brexit party, which is proving attractive to Conservative members.

A survey for the ConservativeHome website on Tuesday suggested three in five Tory members were planning to vote for the Brexit party.

One Tory MEP candidate told the Guardian that the need to campaign was beginning to hit home as the potential gains of the Brexit party became clear.

Labour also faces challenges, after a five-hour wrangle over the precise wording of its manifesto on the circumstances in which it could support a second referendum.