Tory MP Nadine Dorries repeated her call for former Brexit secretary David Davis to replace Mrs May as leader.
The so-called "backstop" will apply if there is no proper EU-UK trade deal post Brexit. But the problem is we can not have a border in Ireland between the Republic and the province [of Northern Ireland].
This week's summit in Brussels has been billed as a make-or-break moment for Brexit talks after a similar meeting last month ended in acrimony.
May urged both parties to show "courage, trust and leadership", but came to Brussels without the concrete new proposals the European Union has asked for.
The move, which would effectively prolong the terms of Britain's European Union membership, could potentially break the impasse. Any deal that is struck will have to be approved by the British and European Parliaments.
Although Barnier said the two sides need "much more time" to work "calmly and patiently" on a deal.
The Prime Minister will briefly address the leaders of the European Union 27 on Wednesday evening before they discuss the state of play in the Brexit negotiations over a working dinner while she leaves.
Mrs May told reporters Thursday morning: "A further idea that has emerged - and it is an idea at this stage - is to create an option to extend the implementation period for a matter of months - and it would only be for a matter of months".
Dutch Prime Minister Mark Rutte said he was "cautiously optimistic" that an agreement would be made "in the coming weeks".
French president Emmanuel Macron said: "We've made a lot of progress but now I think it's time to find an agreement".
In the middle of this year, October was touted to be the date where a deal should come to fruition but around September that timeline was pushed back to November instead. It has made progress more hard rather than less hard.
Despite the lack of progress, the mood music at the summit was positive. A deal must be sealed soon so parliaments have time to give their verdict on it.More news: Westbrook to miss opener vs. Warriors
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"There's no need to dramatize matters". In fact the opposite has been the case and now negotiations are teetering over the edge.
The idea of extending the transition would be to give more time to negotiate a deal on future relations and find a formula to defuse the Irish border question.
But she failed to respond to a call by EU President Donald Tusk for "concrete proposals" of her own to move the talks forward.
Antonio Tajani, the president of the European Parliament who heard the speech, said May offered them "nothing substantially new" to discuss when they retired for dinner afterwards without her.
The issue of Northern Ireland remains a thorny issue in Brexit talks.
The border has been a persistent obstacle in negotiations, and Theresa May has rejected EU suggestions of a backstop which would see the Northern Ireland remain in the EU customs union and bound by the bloc's trade rules.
Britain says it has not asked for an extension - but didn't rule it out Wednesday.
But the two sides disagree on how to resolve the issue.
Reaching agreement with the bloc is only part of the British leader's problem.
Mr McCann's daughter Mary Casey, who was 21 when her father died, spoke of her fear of border custom posts returning in the wake of a no-deal Brexit.
It would mean the United Kingdom remaining within the single market and customs union and subject to EU rules and regulations for nearly three years after the official date of Brexit in March 2019, and more than five years after the referendum vote to leave the bloc.
A diplomat told the Financial Times: 'The extension is an example of how we could be flexible to help the British side if they want it'.