Diplomats say European Union negotiator Michel Barnier is willing to add a year to the already planned 21-month post Brexit transition period, when Britain's relationship with the bloc stays largely the same.
Should a solution not be found, the transition period, which is supposed to run until the end of 2020, could be extended by a year.
Will the proposal solve anything?
"This prolongation of the transition period probably will happen".
"This was the front page of an Irish newspaper published yesterday, interviewing family members, the daughter of somebody who was killed at a customs post on the border between Ireland and Northern Ireland back in the 1970s".
The UK has signed up to the principle of agreeing an Irish border "backstop" - an insurance policy created to prevent the need for customs checks - in case there is a gap between the transition period and the future permanent relationship coming into force. They also impinge on the legally binding Good Friday agreement, which brought peace to Northern Ireland and in some senses pooled sovereignty of Northern Ireland giving people a birthright to be Irish or British or both.
The DUP, who Mrs May relies upon at Westminster to prop up her government, said an extended transition period would mean the United Kingdom continues to "pay but have no say" in Brussels. The Irish have likened it to an insurance policy.
The former Work and Pensions Secretary said: "It is very, very simple we now need to say to the European Union that there is no way on earth we are going to give you any extension".
At a press conference in Brussels, Mrs May said an extension to the transition period "could be a further solution to this issue of the backstop in Northern Ireland". Others fear May is leading the country into a no-deal Brexit, when Britain leaves the bloc without a deal. Britain would have to negotiate this but it has been estimated at anywhere between £10bn and £17bn.
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The EU is now aiming for the December summit to get the deal done, according to one of the officials, though the idea of a November meeting hasn't been ruled out.
A DUP MP has said the prime minister "must be mad" after Theresa May suggested a delay to the UK's departure from the European Union in order to deal with the Irish border issue.
But let me recall that in her Florence speech in September 2017, Prime Minister May proposed a transition period of around two years. May's proposal to extend this period is therefore entirely sensible, even if the realisation that such an extension is needed has come very late in the process.
Meanwhile, Leave campaigner and former UKIP leader Nigel Farage tweeted that "Mrs May's acceptance of an extension to the transition period will take us to the next general election which may mean we never leave at all".
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.
Dutch Prime Minister Mark Rutte, traditionally an ally of May's in the bloc, said "we have the instruments on the table and it should be doable".
May did not offer new ideas to break the deadlock in Brexit negotiations, which have stalled over border issues on the island of Ireland, when she met EU leaders at the start of a summit in Brussels on Wednesday, but showed "will to make headway", said Antonio Tajani, president of the European Parliament.
The proposed 12-month extension would cost billions of pounds in EU fees and leave Britain under European rules for nearly six years after the referendum result.
"What has now emerged is the idea that an option to extend the implementation period could be a further solution to this issue of the backstop in Northern Ireland", she said.
Addressing the summit, Theresa May said there was a "lot of hard work ahead" and "difficult moments" as negotiators enter the final stage of talks.
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