British Foreign Minister Jeremy Hunt warned earlier in the day that progress on the thorny problem of the Irish border is the "only way" to avoid Britain crashing out without a deal on March 29.
She was even pictured sharing a rare laugh with Jean-Claude Juncker - when the EU Commission head showed her his plaster-covered shaving wound on his cheek.
A no-deal Brexit "will have awful economic and social consequences both in Britain and on the continent", Juncker said at a European Union conference, adding he was doing his utmost so "the worst can be avoided".
But unless the United Kingdom can finalise a Withdrawal Agreement by March 29 allowing Britain to continue participating in European Union trade deals during a 21-month transition period, it will enjoy the benefits of the Japanese agreement for just 57 days.
The stalemate has raised fears of Britain leaving without a divorce deal in place at the end of March, an outcome that could cause severe economic disruption.
People's Vote supporter and Labour MP Stephen Doughty said: "Brexiters promised that voting Leave would mean a bonanza of new worldwide trade deals that would make up for lost trade with the EU".
Speaking in Berlin as part of an intense Brexit diplomatic offensive, Foreign Secretary Jeremy Hunt said the key to getting a deal approved by MPs was securing changes which would allow Attorney General Geoffrey Cox to alter his legal advice that the United Kingdom could potentially be trapped in the backstop indefinitely.
A spokesman for May called the Brussels trip "significant" as part of a process of engagement to try to agree on the changes her government says parliament needs to pass the deal.
They say that, among other things, they are not happy with their former parties' Brexit proposals.
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"That is one of the reasons why I think it is so important that we should not leave without a deal".
Diplomats said the two sides were moving toward a separate legal statement in which the bloc would again stress the temporary nature of the backstop for the Irish border.
Brexiteers in May's own Conservative party see this as a "trap" to keep Britain in a form of union indefinitely, and have demanded a time limit or a unilateral exit clause.
"If no deal were to happen, and I can not exclude this, this would have awful economic and social consequences in Britain and on the continent", he said.
But Lord Callanan said: "The Prime Minister has been very clear that we intend to leave on March 29, that's what Article 50 says and that's what our domestic legislation says".
The EU hasn't yet been presented with a possible compromise text by the United Kingdom and the mood on the European side is far gloomier than the one suggested by the British government, two officials said.
Hammond said May's talks with Juncker, had gone well.
Mrs May said Mr Barclay and Attorney General Geoffrey Cox will be in Brussels on Thursday for more talks.
"The ideas for reopening the Withdrawal Agreement are basically for changing the backstop and making it no longer a genuine backstop, because a backstop has to be there whatever the circumstances", Stefaan de Rynck told the United Kingdom in a Changing Europe think tank in London.
The EU has said multiple times it is not willing to renegotiate and that avoiding a hard border between Ireland and Northern Ireland is their priority.
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