One EU diplomat said there was still a "very thin opportunity" to strike a deal, and described a "sense of emergency" in Brussels.
European Union negotiators and leaders have said that Britain should not seek to cherry-pick the best parts of staying in the European Union and leave the tough parts out in its withdrawal agreement and future relations. This year, the government suggested Britain could apply the EU's external tariffs for a limited period in case of any delay in implementing a Brexit deal.
Her counter-proposal, in which all of Britain would remain in the EU's customs sphere, was rejected by the EU.
He said European Union leaders would discuss how to step up preparations for a "no-deal scenario", but stressed that did not mean they were not also making "every effort to reach the best agreement possible for all sides".
There is less than six months to go until the deadline of the 29th of March 2019, when Britain is scheduled to exit the EU.
Mrs May's Brexit plan has also been rejected by leaders of the main opposition Labour party, further dimming the prime minister's hopes of winning parliamentary backing for any Brexit deal she reaches with European Union officials.
Moreover, EU chief Brexit negotiator Michel Barnier said on Sunday that despite intense talks with UK Brexit Secretary Dominic Raab, some key issues of Brexit remain open, including the matter of the Irish border.
As a result, expect today's summit to continue efforts to bridge differences between the two sides, but also to highlight Brussels' continued efforts to prepare for a hard Brexit.More news: Is Meghan Markle traveling with a glam squad?
"There is a Brexit deal that could command the support of Parliament and the country - a Brexit deal that would benefit Britain and allow us to rebuild our communities, regions and economy, and avoid any hard border in Northern Ireland - but that is not her deal", Corbyn said.
Raab's office said: "With several big issues still to resolve, including the Northern Ireland backstop, it was jointly agreed that face-to-face talks were necessary".
But shadow foreign secretary Emily Thornberry told the BBC: "We said we wanted a meaningful vote and we can't see why we should have, on the one hand Theresa May's nonsense and on the other hand a no deal, because that's what they're threatening us with".
Speaking on RTÉ's Morning Ireland, Anna Soubry says the only solution would have been for the United Kingdom to stay in the single market and the customs union, which would have been good for British business and solved the Irish border problem.
She faces a tough reception at the European Council meeting amid demands from President Donald Tusk to bring "creative" proposals to break the impasse over the future of the border in Ireland.
She said it was not just unionists in NI who were determined to fend off the EU's mooted backstop plan, highlighting Scottish Conservative leader Ruth Davidson's opposition to any move that would place a border in the Irish Sea.
The head of a Northern Irish party that props up her government also raised the stakes, saying it was ready to trigger a so-called no deal Brexit to prevent what it called "the annexation" of the British province by the EU.
"But at the same time, responsible as we are, we must prepare the European Union for a no-deal scenario, which is more likely than ever before", he said.