"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.
A Downing Street spokesperson said: "The UK Government regularly engages with the Northern Ireland parties on a range of issues, including on EU Exit, and will continue to offer these engagements".
The prime minister had optimistically claimed that a deal was in reach at the start of her presentation - only to end with a "cry for help" in which she emphasised the difficulties of her predicament at home, one European Union diplomat said.
He also told reporters on the third day of European Union summit meetings in Brussels that it would be up to British Prime Minister Theresa May to decide whether she had to defy opposition from her unionist Northern Irish allies and agree to the EU's proposed backstop, which would potentially create trade barriers between Northern Ireland and the British mainland.
British Prime Minister Theresa May says the remaining differences on Brexit terms between the United Kingdom and the European Union are "few but considerable" - but she says she is confident the two sides will reach an agreement.
Officials said there were hints that Britain might consider extending a post-Brexit transition period beyond the proposed 21 months to make talks on a future trade relationship easier.
EU Council President Donald Tusk said the summit also left him more optimistic than he'd been after a bad-tempered meeting last month and a negotiating crisis on Sunday.
"But I believe everybody around the table wants to get a deal".
Sinn Féin's Michelle O'Neill said the prime minister was "ignoring majority voices" and SDLP leader Colum Eastwood said it was "disgraceful".
European Union leaders have recently suggested that the transition period, now due to end in December 2020, could be extended by a year to provide more time for a trade deal to be forged that would make the backstop unnecessary.
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.
Reaching agreement with the bloc is only part of the British leader's problem.
Mrs May relies on her DUP allies' 10 MPs to prop up her minority government in key votes and there have been suggestions the Northern Irish party could vote against the Budget. "But we are defending the single market".
He says Conservatives are "close to despair" over how the negotiations are going.
But, after Britain's refusal to accept an indefinite legal "backstop" to prevent the return of a hard border between Northern Ireland and the Irish Republic, doubts are mounting.
But time is running out: taking into account British legislative constraints, an agreement has to be reached by late November in order to be concluded and signed on 29 March 2019, the official Brexit date.
"We know how divided British politics is at the moment, not just the Government, and the Government party but also Parliament itself".
"It's up to the prime minister and her teams to come back to the EU", he merely stated at the conclusion of the European Council on Thursday afternoon, where nothing seemed to have happened. "May is a positive political message: we want to achieve an agreement", Tajani said.
The issue of Northern Ireland in these talks is a conundrum that Tusk compared to a "new version of the Gordian knot".
Both sides think an extension would allow for more time to find a solution to the Irish border.