Mrs. May had hoped to lure Labour with the prospect that Britain could stay — at least temporarily — in a type of customs union with the bloc, thereby eliminating the need for tariffs and many border checks on goods flowing between Britain and continental Europe.
But that did not prove enough to tempt Labour, which is in favor or retaining closer ties to the European Union to protect the economy. It had argued that Mrs. May had not offered enough concessions and that, in any case, her successor might be a hard-line Brexit supporter who would tear up any agreement.
Next week Britons will vote in elections to the European Parliament on May 23, almost three years after they voted in a referendum to quit the bloc.
Mrs. May now has to gird herself for the European elections next week hat could prove disastrous for both the Conservatives and Labour. One recent poll has them trailing both the Brexit Party, led by the hard-line Brexiteer Nigel Farage, and the anti-Brexit Liberal Democrats.
The government is expected to try to seek a consensus in Parliament on the way ahead by holding votes on different options, but previous efforts to do so have failed.
Mrs. May wants the Labour Party to agree to abide by the outcome of such votes, but there is little incentive for the opposition to sign up to an outcome that it cannot control. And, without the support of Labour, that process looks doomed to leave Brexit at an impasse.
However, pressure on Mrs. May and Mr. Corbyn to reach a deal increased earlier this month, following local elections that were bad for both parties. While the Conservatives lost 1,300 seats in local municipalities, Labour failed to take advantage, shedding around 80 itself.
Though Mr. Corbyn, a lifelong Euroskeptic, argued that a deal needed to be reached, his party’s divisions have made that outcome difficult. A very large faction in the Labour Party opposes Brexit and wants a second referendum on any deal, so any agreement with the Conservatives would have risked straining loyalty to a breaking point.