The Prime Minister told a crunch European Council summit in Brussels that she was "ready to consider" the extension floated by EU chief Brexit negotiator Michel Barnier to give time to resolve the ongoing row over the Irish border.
If agreed, the change would mean the UK remaining within the single market and customs union and subject to EU rules and regulations for almost three years after the official date of Brexit in March 2019.
And it would be more than five years after the historic referendum vote to Leave.
Brexiteer-in-chief and former UKIP leader Nigel Farage said that it would delay full withdrawal almost until the general election scheduled for May 2022 and "may mean we never leave at all".
After her Chequers plan for Brexit was humiliatingly rejected at the last EU summit in Salzburg last month and efforts to seal a last-minute deal foundered last weekend over the EU's demand for a "backstop" to avoid a hard border in Ireland.
Mrs May has been fighting to keep the door open for an agreement to deliver an orderly withdrawal.
“I remain confident of a good outcome”
Addressing leaders of the 27 remaining EU member states in Brussels, she stressed that significant progress had been made in many areas of the negotiations and urged them to find a "creative" way out of the current dilemma.
The PM said: "We have shown we can do difficult deals together constructively. I remain confident of a good outcome."
And she told them: "The last stage will need courage, trust and leadership on both sides."
Following her 20-minute speech, Mrs May left to eat at the UK ambassador's residence in the Belgian capital, leaving the EU leaders to discuss Brexit in her absence over a dinner of turbot cooked in wheat beer.
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Mrs May's future as Prime Minister appears to be staked on the success of her Brexit negotiations – with infighting within her own party threatening her position.
Brexiteers such as Boris Johnson and Jacob Rees-Mogg have been critical of the PM's handling of Brexit, and there is cabal of 40 Tory MPs who have pledged to vote down her proposals.
Britain continues to be split by Brexit more than two years since the referendum – which was settled in favour of Leave by 52% to 48%.
Thousands expected to march on Westminster this weekend calling for a so-called "People's Vote" on the final Brexit deal reached by the PM.
Labour leader Jeremy Corbyn has slammed Mrs May's approach as "blindfolded" as the oppositions continues to pressure for a new General Election.
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The president of the European Parliament, Antonio Tajani, who also spoke, said: "Both sides mentioned the idea of an extension of the transition period as one possibility which is on the table and would have to be looked into."
A senior EU official later said Mrs May had indicated she was "ready to consider" a longer transition period.
Mrs May initially suggested an "implementation period" of around two years after Brexit, to give the UK's authorities and companies time to prepare for the new arrangements.
But she later accepted a 21-month transition offered by the EU, ending on the last day of December 2020.
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It emerged on Wednesday that Mr Barnier was ready to discuss a further year's extension to allow time to find a solution to keep the Irish border open.
UK officials stressed that the Prime Minister was not proposing any extension to the period already agreed.
Any further extension is likely to be fiercely opposed by Eurosceptics, who warn the UK would become a "vassal state" of Brussels, bound by its rules but unable to influence them.
And it is certain to involve a demand from Brussels for a further year's contributions towards EU budgets, which could cost the UK as much as £9 billion.
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This week's summit has long been billed as "the moment of truth" when agreement was needed to allow time for ratification before Brexit day in March.
But Mr Barnier made clear no breakthrough was now expected, saying that "much more time" was needed to bridge differences between the two sides, and promising to "continue the work in the next weeks calmly and patiently".
Mrs May did not come forward with the new "concrete proposals" on the border issue which European Council president Donald Tusk has said are needed to break the deadlock.
A number of EU leaders voiced their willingness to work for an orderly UK withdrawal – but several also noted that their countries were beginning preparations for a possible no-deal Brexit.