The UK is now due to leave the European Union on 29 March - with or without a deal.
"That's why, in line with our conference policy, we are committed to also putting forward or supporting an amendment in favour of a public vote to prevent a damaging Tory Brexit being forced on the country".
She didn't just say once that she would never countenance a delay in Brexit, and that Britain will definitely leave on 29 March.
He called on May to rule out a "reckless, cliff-edge" Brexit.
"I've no doubt we'll be in legislative mode in the event of a hard Brexit not only for months but for years to come".
Dutch Prime Minister Mark Rutte warned her against "sleepwalking" into a chaotic Brexit next month.
"Let me be clear, I don't want to see Article 50 extended", May said.
Ministers had hoped to have all 40 deals in place by the Brexit deadline, but so far just six have been signed and they don't include large trading partners such as Canada, Japan or South Korea. "It is not an actual alternative course of action in its own right", he told Today.
Her acceptance of a possible extension indicates how much she and her weak Government have ceded control over Brexit to an assertive but divided Parliament.
The shift in policy, which would see voters asked to decide between a deal and remaining in the European Union, won plaudits from Remain-supporting MPs but led to warnings of electoral disaster in some of Labour's heartlands.
Shadow Brexit secretary Sir Keir Starmer confirmed that if Theresa May's deal got through Parliament, Labour's policy was for it to be put to a referendum - with remaining in the European Union as the alternative option.
"I believe that in the situation we are in, an extension would be a rational solution", Tusk told reporters at an EU-Arab League summit in Egypt after talks with May that he said included discussions over extending the Brexit process.
If her plans are backed by MPs on Wednesday, Mrs May will return to the Commons by March 12 for a further "meaningful vote".
And Mr Rutte, who described himself as one of the UK's best friends, said they are "sleepwalking" into chaos. "Unfortunately, we're not able to force her on this".
Labour leader Jeremy Corbyn bowed to the pressure from the more EU-friendly wing of his party by offering a way to halt "a damaging Tory Brexit".
Daniel Hannan has warned that if the Labour Party "get their way" on holding a second referendum the results could be "catastrophic".
"I don't think we should have a second referendum". "A second referendum doesn't do that and the voters - in very, very large numbers - will not accept that". Labour previously said it would support a referendum as a last resort if it could not secure a new general election or make changes to May's divorce deal.
Jacob Rees-Mogg, a leading Brexit-supporting lawmaker in May's Conservative Party, expressed concern: "If it's being delayed.as a plot to stop Brexit altogether, then I think that would be the most grievous error that politicians could commit".
Speculation is mounting that lawmakers will in a series of votes this week move to delay Britain's withdrawal to avoid a no-deal Brexit.
To uproar in the Commons, Mrs told MPs: "They are commitments I am making as Prime Minister and I will stick by them, as I have previous commitments to make statements and table amendable motions by specific dates".