"We are working with European Union to deal with this issue of ensuring that if there is a gap between the end of the implementation period and the point when the future relationship comes in - and we don't expect a gap to exist - we want to ensure there's no hard border between Ireland and Northern Ireland, " May said.
Irish Prime Minister Leo Varadkar said on Friday that a possible extension of Britain's post-Brexit transition period would not diminish Ireland's need for a "backstop" to avoid a hard border with Northern Ireland.
Tajani said the European Parliament, which must endorse any Brexit deal sealed between London and the EU for it to take effect, was in favour of extending the post-Brexit transition period to three years from 21 months.
Mr Barnier said there was absolutely no intention by European leaders to punish Britain.
She subsequently said that the United Kingdom had already put forward a proposal to avoid the need for either a hard border or a customs border between Northern Ireland and the rest of the UK.
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Tusk has made it clear that if May and Barnier do not offer signs of concrete progress towards a draft deal he will not call a November summit to sign it.
Following her 20-minute speech, Mrs May left to eat at the United Kingdom ambassador's residence in the Belgian capital, leaving the European Union leaders to discuss Brexit in her absence over a dinner of turbot cooked in wheat beer.
She also faces a rebellion from her parliamentary partners, Northern Ireland's Democratic Unionist Party, which has threatened to vote against her government's budget if she concedes to the EU's demands over the backstop.
Britain leaves the European Union on March 29, but a deal must be sealed soon so parliaments have time to give their verdict.
"Until now, May has insisted that a legal guarantee to ensure no new border emerges on the island of Ireland should be strictly limited in time".
This week's summit has always been billed as "the moment of truth" when agreement was needed to allow time for ratification before Brexit day in March.
And one person familiar with the discussions said May's Tory party would find it hard to be fighting the next general election - due in 2022 - while the country is still inside the single market and customs union.
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Earlier, May had urged her fellow leaders to work with her to find a "creative way out of this dilemma" on Ireland, a British official said.
The parliamentary vote, and May's political struggles, were the focus of much of the other European Union leaders' dinner late on Wednesday, an aide to French President Emmanuel Macron said.
More positive noises out of Brussels about reaching a Brexit agreement - the dominant driver of sterling - have been offset by growing unease within Prime Minister Theresa May's Conservative party about the sort of deal she is hoping to forge.
Mrs May, who had been called upon to table new proposals on the Irish Border backstop, was thought to have been "developing her thinking".
Extending the transition period could mean that if a future partnership is not ready, a backstop, which so far has been unpalatable to the British side, would not have to be triggered.
Her critics have the numbers to do this but say they have held off so far because they believe she would survive.
Both sides think an extension would allow for more time to find a solution to the Irish border. "If they knew the negotiations were being pushed right up against the next election, why would they be conciliatory in these negotiations?"
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