May is due to meet European Commission chief Jean-Claude Juncker at 1730 GMT on Wednesday, two days after Brexit Secretary Stephen Barclay and Attorney General Geoffrey Cox visited the EU headquarters.
May's spokesperson said the prime minister was "working hard to secure legally binding changes" to the backstop.
Brexit-driven political cracks yawned wider Wednesday as three pro-European lawmakers quit May's ruling Conservatives to join a new centrist group of independents who oppose May's determination to take Britain out of the European Union with or without a divorce deal.
"While the commission engaged seriously with these proposals, it expressed concerns about their viability to resolve the backstop", a DExEU spokesman said.
Brexiteers in May's Conservative party see the clause as a trap that would keep Britain tied to the union indefinitely and have demanded a time limit or mechanism that would allow it to leave the customs union unilaterally.
Juncker on Monday gave the EU's clearest signal yet that London could seek a long delay of its exit date of March 29.
Downing Street said it expected the prime minister and Mr Juncker to have detailed discussions about the shape of any new text - dubbed the "Cox codicil" - that would satisfy MPs.
"If you can get to a place where the potential longevity of the backstop - the potential that the backstop lasts for ever - can be adequately dealt with, that's what we are all seeking to do", he said.More news: Jihadi bride Shamima Begum claims Manchester bomb was justified
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He added that an orderly departure was "of paramount importance", and that pushing back the deadline would leave both the United Kingdom and European Union in "paralysis".
If Britain leaves the European Union with no deal on trade, border and other issues, it faces risks of a recession that may reach the scale it experienced in the early 1990s when Britain's gross domestic product fell by 2 percent over six quarters, the rating agency said.
The EU refuses to budge, at least as far as changing the 585-page legally binding Brexit agreement is concerned.
Manufacturing supply chains could be disrupted, and Brexit uncertainty has already been cited as a contributing factor in the closure or departure of several British-based businesses.
Revenue said over three times more companies applied in the month of February than in January for Economic Operators Registration and Identification (EORI) numbers.
Avoidance of a disruptive Brexit, potentially by the implementation of a withdrawal agreement, thereby ensuring a smooth Brexit transition process.
Member states are obliged to hold polls between 23 and 26 May to elect a new European Parliament, which will start sitting from 2 July, presumably without any British members. "If such a request were to be made, no one in Europe would oppose it", Juncker told the Stuttgarter Nachrichten this week.
The Labour leader Jeremy Corbyn is to try to break the Brexit deadlock.