The Prime Minister’s official spokesman told journalists that Mrs May had warned MPs ahead of the second vote on her deal that the UK would be plunged into crisis if they failed to back it, adding: “That has come to pass”.
Downing Street confirmed that a letter would be sent to the EU Council President Donald Tusk on Tuesday or Wednesday, setting out the UK’s request for an extension to Article 50 to delay Brexit, in the absence of an agreed deal.
The request will be discussed by EU leaders at a summit in Brussels on Thursday.
Reports suggested the Prime Minister did not state a preference to cabinet, a government source denied this.
Mrs May’s spokesman said she voiced her "absolute determination" that MPs should have another chance to vote on her Brexit deal, despite the bombshell intervention of the Commons Speaker. But the meeting is said to have descended into a row, with the Leader of the Commons, Andrea Leadsom accusing colleagues of failing to deliver Brexit.
John Bercow provoked uproar at Westminster on Monday when he ruled that the Government could not bring the Prime Minister's deal back for a third "meaningful vote" unless there were substantial changes.
In the course of a 90-minute discussion with senior ministers, Mrs May made clear she wanted MPs to have another vote "as soon as possible".
The Prime Minister's official spokesman said: "What you can see from the Prime Minister and her colleagues is an absolute determination to find a way in which Parliament could vote for the UK to leave the European Union with a deal.
"The Prime Minister has been very clear throughout that she wants that to happen as soon as possible."
The spokesman added: "She has said in the House of Commons that she does not want there to be a long delay and that she believes asking the British public to take part in European elections three years after they voted to leave the EU would represent a failure by politicians."
Earlier, Brexit Secretary Stephen Barclay signalled that ministers would press on with Mrs May's Brexit deal despite the Commons Speaker's bombshell intervention.
Mr Barclay said that, while the Cabinet would give "serious consideration" to his ruling, Mrs May's plan remained "the only deal on the table".
"What we need to do is secure the deal," he told Sky News. "This is the only deal on the table. The EU is clear it is the only deal on the table. Business need the certainty of this deal and it is time that Parliament comes together and gets behind it."
Mr Barclay acknowledged that the ruling made it "more unlikely" that there would be an attempt to get the deal through the Commons before Mrs May attends the EU summit in Brussels on Thursday.
However, he said the Speaker had made clear in earlier rulings that the Commons should not necessarily be bound by precedent.
"What the Speaker has said in his ruling is there needs to be something that is different. You can have the same motion but where the circumstances have changed," he said.
"So we need to look at the details of the ruling, we need to consider that in the terms of earlier rulings that don't particularly align with yesterday's.
Mr Barclay rejected suggestions that ministers could seek a prorogation of Parliament from the Queen in order to get round the ruling.
"I think the one thing everyone would agree on is involving Her Majesty in any of the issues around Brexit is not the way forward so I don't see that as a realistic option," he said.
He accepted that there would now have to be a "short extension" to the Article 50 withdrawal process as the Government would not be able to get through all the legislation it needed in time for March 29 when the UK is due to leave.