The EU meeting is the occasion when the leaders of the remaining 27 member states were supposed to give the green light for a special summit in November to finalise the terms of Britain's withdrawal.
The Prime Minister brought up the possibility of an extension during meetings with European Union leaders in Brussels on Wednesday in an effort to get passed a deadlock in negotiations.
The 27 will then dine without May and hear reports from Barnier and from European Commission President Jean-Claude Juncker.
The proposed extension would buy the United Kingdom more time to agree a trade deal with the bloc, which could then reduce the likelihood that a backstop would need to be implemented in the first place.
But when the prime minister was asked in the House of Commons earlier Wednesday whether her government's blueprint for an amicable divorce was dead, May replied: "The answer is no". He added, "for a breakthrough to take place, besides goodwill, we need new facts".
May earlier told parliament in London she wanted any treaty that includes the Irish border issue linked to a separate declaration the two sides intend to issue on their intentions for a close EU-UK trade pact to be negotiated after Brexit.
But three days after talks stalled over the Irish border "backstop", thwarting hopes of a deal at the summit, May arrived determined to stress that a deal was still on the cards.
He suggested it was not clear a year ago what agreeing to set up a so-called backstop on the Irish border would mean - effectively accusing civil servants of underplaying the issue.
German Chancellor Angela Merkel said Wednesday that she still believes it's possible to conclude a "good and sustainable" agreement with Britain before its departure - but stressed Germany also is preparing for the risk of a no-deal departure.
The summit also was meant to lay the groundwork for an additional planning meeting next month for the U.K.'s exit - though, as the BBC writes, it's possible that European Union leaders will formally say they are going to use the November gathering to prepare for that no-deal scenario.
SNP Westminster leader Ian Blackford later urged the Prime Minister to negotiate continued membership of the single market and customs union to "avoid an economic catastrophe".
Sammy Wilson MP has said Boris Johnson's father Stanley is a loose cannon and a nonentity after he claimed Margaret Thatcher would have fixed the Irish border Brexit dilemma by saying "if the Irish want to shoot each other, they will shoot each other".
Such an extension, keeping Britain under European Union governance with no say in it, would be highly unpopular with hardline supporters of Brexit.
The European Union is ready to extend Brexit's post-divorce transition period by a year to allow more time to find a trade deal, diplomats said Wednesday ahead of a hard summit.
The EU's chief Brexit negotiator, Michel Barnier, warned at the start of the EU leaders summit in Brussels that the Brexit negotiators needed "much more time" to complete a deal.
The "backstop" refers to a commitment from both sides that even if Britain exits without a comprehensive deal, there will never be a hard border between Northern Ireland, which would leave as part of the United Kingdom, and the Republic of Ireland, which will remain part of the European Union.
"Today we do not know what they want", she said.
French president Emmanuel Macron said: "We've made a lot of progress but now I think it's time to find an agreement".