The backstop is a proposed mechanism to keep the United Kingdom in the EU's Customs Union after Brexit - essentially precluding the country from taking back control of its worldwide trade policy - in order to prevent a hard economic border between the British province of Northern Ireland and EU member the Republic of Ireland.
Last December, EU and UK negotiators agreed to a 21 month transition period after the UK formally leaves the bloc in March next year during which the UK would remain part of the single market but have no say over new rules governing it, and on the need to agree a backstop on the Irish border.
European Parliament President Antonio Tajani confirmed that both sides had mooted the idea of extending the transition period by an additional year.
The prime minister and her team are willing to consider the longer transition as a way to help overcome the Irish border issue.
The summit continues Thursday with an agenda limited to some issues both sides firmly agree on, including fighting cybercrime and dealing with an assertive Russian Federation.
"I met the leaders of both of the two main parties last week and they were in full agreement that the uncertainty around Brexit was one of the major reasons why they haven't been able to form an executive".
He went on: 'But, even more important, Labour government has a plan for a post-Brexit Britain where we use the powers available to kick start the economy and rebuild our industry, infrastructure and public services in all regions and nations of the United Kingdom'.
"At the very beginning she threw away the one negotiating lever that she had, namely the £39 billion that they (the EU) want to get their hands on", Mr Wilson said. May spoke of "working intensively over the next days and weeks" to achieve agreement that avoids a no-deal departure from the bloc for Britain on March 29 that could create chaos at the borders and in the economy.
"What has now emerged is the idea that an option to extend the implementation period could be a further solution to this issue of backstop in Northern Ireland".
"It is precisely one of those things that needs to be teased out and addressed, whether we felt and the European Union felt it was the right approach to take".
"People who have been supportive of her throughout this process, they are close to despair at the state of this negotiation", said Conservative lawmaker Nick Boles, a critic of May who during the referendum backed remaining in the European Union and now wants to stay in the single market on an interim basis.
Tusk has made it clear that if May and Barnier do not signal concrete progress towards a draft deal he will not call the planned November summit.
But the meeting that had been painted as a make-or-break moment for a Brexit deal ended with a whimper.
"It is always the case in negotiations that they are tense and hard and challenging at the end".
It's also thought that May agreed that there can not be a time limit on the Northern Ireland backstop, as agreed last December as an insurance policy.
For many who campaigned to leave the European Union, the idea of continuing to pay into the bloc well beyond Britain leaves in March next year is anathema.
As EU leaders gathered yet again in Brussels yesterday, they conceded that hopes of a deal are again being pushed back, this time perhaps until December.
EU Brexit negotiator Michel Barnier sought to calm the mood, saying the only remedy was to use the remaining period between now and Christmas.
It will take place at the same time as a similar rally in London to demand a People's Vote on the terms of the Brexit Withdrawal Agreement. "If they knew the negotiations were being pushed right up against the next election, why would they be conciliatory in these negotiations?" "That is the problem".