Pro-EU lawmakers in Parliament aimed to seize control of the process with a motion later today that would let them hold a series of votes on alternatives to May's deal.
They hope these "indicative votes" on options that include a new voter referendum on EU membership and a "soft Brexit" that maintains close economic ties to the bloc can find majority support.
May said the Government would "engage constructively" with the results of the process. But she said she was sceptical that it would produce a decisive result.
With the March 29 Brexit day set almost two years ago days away and the withdrawal agreement lacking Parliament's approval, European leaders seized control of the timetable last week.
There is wide concern that a no-deal departure would be disruptive for the world's biggest trading bloc and deeply damaging for Britain.
The EU granted May's request for a delay, though a shorter one than she sought.
It said if Parliament approves the proposed divorce deal, the UK will leave the EU on May 22. If not, the Government has until April 12 to tell European leaders what it plans to do — leave without a deal, cancel Brexit or chart a new path.
In agreeing to the postponement, European leaders hoped Britain's deadlocked politicians would find a solution to the crisis. But the EU isn't counting on it. The European Commission said today that it had completed planning for a no-deal Brexit, calling that outcome "increasingly likely".
May stands little chance of getting the deal she struck with the EU approved unless she can win over Brexit-backing MPs in her Conservative Party and its Northern Irish ally, the Democratic Unionist Party. The DUP said that the party's "position remains unchanged".
May has come under intense pressure to quit the prime minister's post as the price of winning support for the deal.
At a meeting at the prime minister's country retreat, Chequers, prominent Brexiteers told May they might back the deal — if she agreed to step down so that a new leader could take charge of the next phase of negotiations, which will settle Britain's future relations with the EU.
Former Foreign Secretary Boris Johnson, who attended the meeting, accused the Government of lacking "gumption" and chickening out on delivering Brexit.
Writing in the Daily Telegraph, Johnson said that to win support for her deal, May must show that the next phase of negotiations "will be different from the first". Johnson is likely to be a contender in any future Conservative leadership race.
The Sun piled pressure on May with a front-page call for the Prime Minister to resign, under the headline "Time's up, Theresa."
May is hanging on, hoping she can persuade Brexit-backing MPs that rejecting her deal means Britain may never leave the EU.
She told MPs that Britain would not leave the EU without a deal unless Parliament — which has already rejected the idea — voted for it.
She added that cancelling Brexit "must not happen," while "a slow Brexit" that involved a long delay to Britain's departure, "is not a Brexit that will bring the British people together."
The EU's executive Commission said that it had finished planning for a no-deal Brexit, which could occur on April 12.
The remaining 27 EU countries are hiring hundreds of people to conduct extra border and customs checks as well as to staff call centres set up to handle Brexit inquiries.
The commission warned that despite the preparations, a cliff-edge Brexit would cause "significant disruption for citizens and businesses." It said new tariffs and border checks would cause delays for both people and goods.
The bloc also said that more remained to be done on ensuring an open border between EU member Ireland and Northern Ireland — something both sides have agreed to.
Checkpoints there were a source of great tensions during the "Troubles" from 1968 until the Good Friday peace agreement was sealed in 1998.
An EU official said the bloc was in "in intense discussions with the Irish authorities about these matters."
Opponents of Brexit feel the political tide may be turning in their favour. Hundreds of thousands of people marched through London at the weekend calling for a new referendum on whether to leave the EU or remain.
But with the deadline for a Brexit decision less than three weeks away, British politicians remain divided, and increasingly despairing about the country's political gridlock.
"Brexit is like the Death Star of politics," Conservative legislator George Freeman said. "I always feared it would be like this. It's destroying and soaking up all the Prime Minister's room for manoeuver and political goodwill.
"I've never known this country so divided, so angry and in such a dangerous state," he said.
- AP
