The DUP has accused the Prime Minister of "holding out a begging bowl to European leaders" after she made the case for a delay to Brexit.
Theresa May spent yesterday holding talks with French President Emmanuel Macron in Paris and German Chancellor Angela Merkel in Berlin as she asked for a longer extension to Brexit until June 30.
However, European Council president Donald Tusk has suggested EU leaders grant the UK a longer extension to Brexit than Mrs May has requested.
Mr Tusk said there was "little reason to believe" that the ratification of Mrs May's beleaguered Brexit deal could be completed by the end of June.
He called for the European Council to discuss an alternative, longer extension, such as a "flexible extension" lasting "as long as necessary and no longer than one year".
DUP deputy leader Nigel Dodds MP said the negotiation between the Prime Minister and the leaders of France and Germany "is humiliating and embarrassing for the United Kingdom".
"The problems the Prime Minister is attempting to solve were not created by the decision to leave the European Union, rather the ineffective negotiations by the Prime Minister to implement that decision," he claimed. "A solution was put forward by the House of Commons on 29th January through the Brady amendment, which could have charted a course by which Parliament could have supported a deal.
"Instead of treating this proposal seriously, the Prime Minister has effectively accepted the backstop and opted not to stand up to Brussels.
"Nearly three years after the referendum the UK is today effectively holding out a begging bowl to European leaders."
Mrs May has asked for the date of Brexit to be delayed until June 30 at today's crunch summit, with the possibility of an earlier departure if the UK's withdrawal deal is ratified.
MPs backed her call in a Commons vote yesterday afternoon as they approved a Government motion for Mrs May to seek an extension to June 30 by 420 votes to 110, a majority of 310.
However, 97 Conservatives rebelled by opposing the plan, including former Brexit Secretaries David Davis and Dominic Raab.
But Mr Tusk called for the European Council to discuss an alternative, longer extension, such as a "flexible extension" lasting "as long as necessary and no longer than one year".
Mr Tusk wrote in a letter to the heads of the 27 remaining member states ahead of today's summit: "The flexibility would allow to terminate the extension automatically, as soon as both sides have ratified the withdrawal agreement.
"The UK would be free to leave whenever it is ready. And the EU27 would avoid repeated Brexit summits. Importantly, a long extension would provide more certainty and predictability by removing the threat of constantly shifting cliff-edge dates.
"Furthermore, in the event of a continued stalemate, such a longer extension would allow the UK to rethink its Brexit strategy."
He suggested that the EU would grant an extension rather than allowing Britain to leave without a deal on Friday, saying that, given the "risks posed" for those on both sides of the English Channel, "I trust that we will continue to do our utmost to avoid this scenario".
Mr Tusk's letter came after the PM arrived in Paris for talks with Mr Macron, who in recent days has warned that an extension to the Article 50 withdrawal process is not guaranteed.
Earlier, over a working lunch, Mrs May and Mrs Merkel had agreed on the importance of an "orderly withdrawal" from the EU, Downing Street said.
The unanimous agreement of all 27 remaining EU states is needed to avoid a no-deal Brexit on the scheduled date of April 12.
The visits came as the EU's chief Brexit negotiator Michel Barnier said Brussels could amend the political declaration on future relations with the UK "within a few hours or days" to incorporate the customs union arrangement being discussed in cross-party talks between the Government and Labour.
But there were signs of resistance in Mrs May's Cabinet to compromise with Labour, with International Trade Secretary Liam Fox warning that a customs union would leave the UK "stuck in the worst of both worlds".
Meanwhile, the cross-party talks seeking to break the Brexit impasse will resume tomorrow, with Labour saying the Government had not yet made a "clear shift" in its position.