British MPs have rejected the text, and May's government is now in talks with the opposition Labour party to try to find a way through the deadlock
British Prime Minister Theresa May held talks with German Chancellor Angela Merkel Tuesday in a last-gasp bid to keep her country from crashing out of the European Union at the end of the week.
May will also meet French President Emmanuel Macron in Paris, seeking support from the bloc's powerbrokers for her request to delay Brexit for a second time.
She is hoping EU leaders will agree at an emergency summit on Wednesday to postpone Brexit from April 12 to June 30, to give her more time to get her divorce deal through parliament.
British MPs have rejected the text, and May's government is now in talks with the opposition Labour party to try to find a way through the deadlock.
But these discussions are moving slowly, and EU negotiator Michel Barnier said May must explain in Brussels what another postponement would achieve.
"The length of the extension must be linked to the purpose - what it's for - and that depends on what Mrs May will say to European leaders tomorrow," he told reporters after a meeting of EU ministers in Luxembourg.
He added: "It is for her to bring the roadmap."
The EU has already granted one delay - the original Brexit deadline was March 29 - and Barnier said it would not be the bloc's fault if Britain crashed out this week, risking huge economic disruption.
"No deal will never be the EU's decision, it will always be the responsibility of the United Kingdom to tell us what it wants," he said.
EU ministers have expressed frustration at the turmoil in London, where MPs still cannot decide how to leave the EU almost three years after the referendum vote for Brexit.
"We are in a very, very frustrating situation here," said Germany's Europe minister Michael Roth as he arrived in Luxembourg.
His French counterpart Amelie de Montchalin told reporters that "we want to understand what the UK needs this extension for".
"And then comes the question of the conditions of what role we'd want the UK to play during this extension time," she added.
Britain is requesting only a short delay to avoid having to take part in European Parliament elections next month, but EU leaders are expected to offer a longer postponement.
EU Council president Donald Tusk's office has floated the idea of a "flexible" extension of up to a year, with an option for London to leave earlier if it finds a way through, but there is no agreement on this.
Some in the EU are worried that during a long delay, British representatives could disrupt EU budget planning and reforms during indefinite Brexit talks.
A spokesman for May said Britain had "engaged constructively through this process and you can expect the UK to continue to do so."
Neither May nor Merkel spoke to the media during their talks in Berlin, but the German leader has previously said she would work "until the last hour" to avoid a "no deal" Brexit.
However, Macron - who May meets in Paris on Tuesday evening - has raised the possibility of rejecting the delay request, warning he does not want to simply prolong the uncertainty.
May is now hoping for more progress with Labour, although its leader Jeremy Corbyn said Monday he was still waiting for signs of compromise.
"The problem is that the government doesn't seem to be moving off the original red lines," he said.
His top team was meeting Tuesday with some of May's senior ministers, including leading Brexit supporter Michael Gove and her more pro-European finance minister Philip Hammond.
Corbyn wants Britain to agree a new customs arrangements with the EU after Brexit, and believes these could be included in a text May has already agreed with Brussels setting out the shape of future trade ties.
But many of her ministers and MPs are strongly opposed to the idea of closer ties with the bloc, saying it would undermine the potential of Brexit.