Brexit talks between the Conservatives and Labour are about to close after six weeks, without an agreement.
Theresa May and Jeremy Corbyn will now discuss how to put a series of options to Parliament, designed to find a consensus among MPs for a way forward.
On Thursday Mrs May promised to set a timetable for leaving Downing Street following the next Brexit vote in June.
Ex-Foreign Secretary Boris Johnson is the latest MP to say he will run in the Tory leadership election to follow.
The UK was due to leave the EU on 29 March but the deadline was pushed back to 31 October after MPs rejected Mrs May’s proposed deal – the withdrawal agreement negotiated with the EU – three times.
That prompted attempts to find a way to end the impasse through cross-party talks between Labour and the Conservatives.
But BBC Newsnight political editor Nicholas Watt said Tory whips had given up hope of finding agreement with the Labour leader on a Brexit deal.
He said on the other side of the negotiating table “Labour has fears about the durability of a deal agreed with a weak prime minister”.
At a shadow cabinet meeting on Tuesday, some Labour frontbenchers called for an immediate halt to the talks, raising questions over whether Mr Corbyn could win approval from his party for any deal.
Labour negotiators have been seeking a permanent and comprehensive customs union with the EU after Brexit, meaning that there would be no internal tariffs (taxes) on goods sold between the UK and the rest of the bloc.
But many Brexit-supporting Tory MPs want the UK to negotiate its own trade deals on goods with other countries around the world, which would be impossible with a customs union in place.
The prospect of compromising on issues such as this in the prime minister’s talks with Labour has provoked anger in the Conservative parliamentary party.
Mrs May has come under increasing pressure to leave Downing Street this summer, with Brexit deadlocked and poor results for the Conservatives in the recent local elections in England.
She will try once again to gain the support of MPs in the week beginning 3 June, when the Commons votes for the first time on the EU Withdrawal Agreement Bill – legislation needed to implement her deal.
BBC political editor Laura Kuenssberg said if the bill were defeated Mrs May would be expected to announce that she was stepping down.
Brexiteer Mr Johnson said he would stand in the leadership contest after Mrs May departs, telling a business conference in Manchester on Wednesday: “Of course I am going to go for it.”
Labour backbench MP Hilary Benn, who chairs the Brexit select committee in the House of Commons, said news that the talks were ending had “not come as a great surprise” as it did not appear much progress had been made.
He told BBC Radio 4’s Today programme he believed Labour should back a public vote on any Brexit deal.
“There are only two ways out of the Brexit crisis that we’ve got – either Parliament agrees a deal or we go back to the British people and ask them to make the choice,” Mr Benn said.
“I think this brings the prospect of a confirmatory referendum closer, although there’s not yet a majority for that in Parliament.”
Scotland’s First Minister Nicola Sturgeon told Today that she felt “a degree of sympathy” for Mrs May on a “personal level”.
But the SNP leader added: “That said, I don’t think she has played the hand she was given particularly well.”