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 United Kingdom to leave the European Union with a deal".
Possible eventual outcomes still range from a long postponement, leaving with May's deal, a disruptive exit without a deal, or even another referendum.
Explaining that Mrs May "won't be asking for a long extension" when she writes to the EU, Number 10 said: "There is a case for giving Parliament a bit more time to agree a way forward, but the people of this country have been waiting almost three years now".
So long as Britain remains in the club, it has a voice in its decision-making - an oddity that means that May will also sign off on a raft of non-Brexit EU policies at the summit this week.
But the spokesman said: "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".
This morning, the Tánaiste Simon Coveney travelled to Brussels to meet with EU Brexit Coordinator Guy Verhofstadt and Chief EU Negotiator Michel Barnier.
The BBC's political editor, Laura Kuenssberg, said May would ask for an extension until June 30 - which could give her another chance to pass her deal - with the option of a delay of up to two years.
In another extraordinary day yesterday, the Speaker of the House of Commons, John Bercow, invoked the "Erskine May" parliamentary rules of procedure.
"This is a moment of crisis for our country", he said.
Johnson, a prominent Brexit campaigner who might influence other lawmakers on which way to vote over May's deal, asked in his column in the Telegraph newspaper whether there was a way forward to break the impasse of Brexit in parliament.
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City came back from two goals down at the Liberty Stadium to book a semi-final spot for the second time in three seasons and keep alive their hopes of an unprecedented Quadruple in English football.
"I will fight to the last hour of the deadline on 29 March for an orderly exit", she told a press conference in Berlin.
"The question people like me will ultimately have to answer is: can we get to no-deal instead?"
If there is no extension then the only remaining options would be a no-deal exit or cancelling Brexit.
After two-and-a-half years of tortuous negotiations with the European Union, the final outcome remains uncertain - with options including a long delay, exiting with May's deal, a disorderly exit without a deal or even another European Union membership referendum. It would face the prospect of manufacturing and financial market disruption, sharp economic contraction and border delays.
But Britons' patience with negotiations may be running out. In a Comres survey in the Telegraph newspaper, almost half of respondents said Britain would ultimately thrive if it left without a deal.
"The fact a number of Members of Parliament have said that they will change their votes points to the fact that there are things that are different".
To compound her problems, May appeared unlikely to reach agreement this week on her Brexit plans with the Democratic Unionists, the small Northern Irish party whose support is vital if she is to get her European Union departure deal through parliament.
Delivering his verdict, Bercow said the precedent for stopping the government repeatedly bringing the same motion before MPs dated back to 1604.
She needs to win over at least 75 lawmakers - dozens of rebels in her own Conservative Party, some Labour lawmakers, and the Northern Irish Democratic Unionist Party (DUP), which props up her minority government.
The government had expected to ask for a short extension if the deal had been approved by Parliament, ending before July 2 when the new MEPs take their seats.