BRUSSELS — If Britain ever wants an orderly exit from the European Union, it must pass the withdrawal agreement painfully negotiated with the bloc, Michel Barnier, its chief Brexit negotiator, warned on Tuesday.

Given how little time remains before April 12, when Britain might have to leave the European Union without a deal, Prime Minister Theresa May emerged from a series of tense cabinet meetings to say that she will ask her European counterparts to grant Britain another short extension to try to pass a withdrawal agreement. Parliament has already rejected her plan three times by margins varying from historic to hefty.

A majority in Parliament opposes a no-deal Brexit, but agrees on little else.

To show the Europeans that the extension is worth providing, Mrs. May announced that she would try to meet with the head of the opposition Labour Party, Jeremy Corbyn, “to agree a plan that we would both stick to ensure that we leave the European Union and that we do so with a deal.”

Mr. Corbyn favors a much closer trading relationship in the future with the European Union than Mrs. May. But if the two leaders can agree on a softer Brexit, then a parliamentary majority might be found to leave the European Union with a deal, initiating a transition period of at least two years while a future relationship is negotiated.