Insisting that a Brexit divorce deal was "achievable", May said the border dispute should not "derail the prospects of a good deal and leave us with the no-deal outcome that no one wants".
"I am going to ask Prime Minister May whether she has concrete proposals on how to break the impasse".
Any deal that she manages to negotiate with the European Union needs to be voted on by parliament in order to be legally adopted, some that, in a hung parliament, would be tricky at the best.
Some EU leaders, notably President Emmanuel Macron of France which faces questions over intensive cross-Channel trade and traffic, want much more public clarity to show that the bloc could cope with cutting Britain out of EU regulations.
Brexit talks are on a knife-edge, with British Prime Minister Theresa May set to make a pitch to European Union leaders in Brussels on Wednesday at a summit billed as the last chance to agree a draft deal in time for Brexit day on March 29.
With less than six months to go before Britain's departure, talks stalled at the weekend over how to ensure there is no return to a hard border between the British province of Northern Ireland and EU-member Ireland.
"We are committed to concluding these negotiations with timely and substantive results for American workers, farmers, ranchers, and businesses", Lighthitzer said, a hint that the United States would expect a wide-ranging deal to cover controversial agriculture products.
Multiple reports suggest Cabinet resignations have been put on hold for at least another day with restless ministers prepared to give Prime Minister May more time to satisfy their demands.
Britain insists such a solution can not be permanent, while the European Union says just as strongly that it must not have an end date.
"But we are equally clear that we can not support an outcome that creates a border in the Irish Sea and which divides NI from the rest of the UK".
Turner homers as Dodgers beat Brewers 4-3 in NLCS Game 2
High-powered Los Angeles had just two singles off starter Wade Miley before breaking through against Milwaukee's vaunted bullpen. He was the NLCS co-MVP past year with Chris Taylor when Los Angeles made it to the World Series for the first time since 1988.
Yesterday, Mrs May appealed for her Cabinet to "stand together and stand firm" at a marathon two-and-a-half hour meeting in Downing Street.
Sky News also understands several ministers gathered in Ms Leadsom's office on Monday evening for "pizza club" to discuss resignations and support for the prime minister over takeaway.
Mr Barnier also declined comment after the breakdown, only telling Sky News: "Be patient".
"The EU side is willing to allow more time in the transition period to agree an alternative solution to the backstop", he told BBC Radio 4's Today programme.
Mr Hunt last night used an interview with the Mail to warn that chances of a Brexit deal would be scuppered if ministers failed to get behind Mrs May at this "pivotal moment".
However, he said, the backstop would not work as an insurance option if it was time-limited.
"We sometimes feel the British are negotiating with the British", Belgian Foreign Minister Didier Reynders said, illustrating the EU's frustration.
"Take responsibility and be constructive", said Germany's Europe Minister Michael Roth when he arrived for talks with Barnier in Luxembourg.
Irish Foreign Minister Simon Coveney admitted to being "frustrated" by the delay, saying that apart from Britain, Ireland is the country with most to lose from Brexit. "There are a lot of inaccurate macho talk about countries being able to survive a no-deal Brexit ... the truth is a no deal Brexit creates significant problems".