After sending a formal request to EU Council President Donald Tusk ahead of a summit in Brussels on Thursday, Mrs May told MPs: "As Prime Minister, I am not prepared to delay Brexit any further than June 30."
Asked if that meant she would quit rather than accept a longer extension to Article 50, a Downing Street spokesman said: "You should infer from those words the strength of the prime minister's resolve.”
But with some EU leaders believed to be against any extension, and Brussels opposed to granting a delay without a clear purpose, Mrs May’s letter has also raised renewed fears about the possibility of a no-deal Brexit, as early as the scheduled exit day at the end of next week.
Brussels has made clear that any extension of the Article 50 negotiation process beyond June would require the UK to take part in elections to the European Parliament in May - something which Mrs May said was in the interests neither of Britain or the EU.
She told MPs at Prime Minister's Questions: "The idea that three years after voting to leave the EU, the people of this country should be asked to elect a new set of MEPs is, I believe, unacceptable.
"It would be a failure to deliver on the referendum decision this House said it would deliver."
European Commission president Jean-Claude Juncker said an extension is unlikely to be agreed in Brussels tomorrow, warning there will "probably" have to be another summit next week if Mrs May is unable to show she has the support of Parliament for her deal.
He told German radio that a short extension would give the UK Parliament time to agree to the existing Withdrawal Agreement text, but insisted there would be "no more negotiations".
"If that doesn't happen, and if Great Britain does not leave at the end of March, then we are, I am sorry to say, in the hands of God," said Mr Juncker. "And I think even God sometimes reaches a limit to his patience."
An unconfirmed report in French news magazine Le Point suggested that President Emmanuel Macron will argue against any postponement beyond March 29.
Only six days ago, Mrs May's effective deputy David Lidington warned that a short delay would be "downright reckless". And a motion tabled by the Government last week appeared to keep a longer extension of nine months to a year on the table.
But Mrs May - who spoke with Mr Juncker by phone before facing MPs - told the Commons: "The Government intends to bring forward proposals for a third meaningful vote.
"If that vote is passed, the extension will give the House time to consider the Withdrawal Agreement Bill. If not, the House will have to decide how to proceed.
"But as Prime Minister, I am not prepared to delay Brexit any further than June 30."
If a delay is approved by EU leaders, Mrs May will rush legislation through both Houses of Parliament next week to remove the date March 29 from Brexit laws.
Under EU rules, participation in the European Parliament elections must be confirmed by April 11, creating a hard deadline beyond which a longer extension would become impossible.
SNP MP Pete Wishart told the Prime Minister she was “weak, weak, weak” for bowing to pressure from Brexiteers in her Cabinet and party by only asking for a short extension to Article 50.
And Labour leader Jeremy Corbyn accused Mrs May of "running down the clock" on Brexit.
He told the Commons: "This Government has led the country and themselves into crisis, chaos and division.
"We are still legally due to leave the European Union in nine days' time. Months of running down the clock and a concerted campaign of blackmail, bullying and bribery has failed to convince the House or the country that her deal is anything but a damaging national failure and should be rejected.
"If the Prime Minister cannot get changes to her deal, will she give the people a chance to reject the deal and change the government?"
Tory Brexiteer Peter Bone warned Mrs May she will be "betraying" the public if she continues to seek to delay Brexit.
In her letter to Mr Tusk, Mrs May said Commons Speaker John Bercow ruling on Monday that he would not allow the same motion to be brought again during this session of Parliament unless it was substantially changed made it "impossible" to hold a third meaningful vote ahead of Thursday's summit.
But she indicated she believes it is possible to satisfy the Speaker's requirements by ensuring the European Council formally approves legal assurances on the Irish border backstop agreed last week with Mr Juncker in Strasbourg.