Brexit secretary Steve Barclay and attorney general Geoffrey Cox had separate talks with Mr Barnier.
The economic partnership agreement between Japan and the European Union will cease to apply to the United Kingdom if it exits the EU at the end of March without a deal, the United Kingdom government revealed Thursday.
But EU leaders insist the withdrawal agreement, which they struck with May at a summit past year, cannot now be re-opened, even to appease British lawmakers.
"The simplest way to get legally binding changes on the backstop is to reopen the Withdrawal Agreement".
Downing Street said "productive" talks in Brussels aimed at addressing the MPs' concerns were continuing "urgently".
The government is yet to win parliament's backing for its Brexit deal with Brussels, and if Britain leaves without a deal those bilateral agreements will cease to apply, likely disrupting trade.
However, that parliamentary arithmetic remains precarious with a number of factions unhappy with the government's Brexit strategy.
The Irish government has published legislation created to ease the damage if Britain leaves the European Union without a deal - but says it hopes the law will never be needed.
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However, remain-backing politicians were keen to air their warnings about the dangers that Brexit presented for British business. The UK has yet to seal a deal laying out the divorce terms and establishing what trade rules will apply after Brexit.
A second diplomat, briefed on the May-Juncker talks, confirmed the European Union would only signal this was the direction of travel, rather than offering a precise text, before May faces another vote in parliament.
May will have a further chance to lobby EU leaders at the weekend in the Egyptian resort of Sharm el-Sheikh during an EU-Arab League summit focused on migration, trade and security.
Discussions between the two sides suggest May will ask for an extension to the two-year negotiating period if Parliament backs the Brexit deal but it isn't signed off until an European Union summit on March 21 and 22.
"It is time that many of our Conservative party colleagues in the ERG recognise that Parliament will stop a disastrous no-deal Brexit on March 29".
British lawmakers voted 432-202 against her deal on January 15, the worst defeat in modern British parliamentary history.
"However, if there is no breakthrough in the coming week, the balance of opinion in Parliament is clear - that it would be better to seek to extend Article 50 and delay our date of departure rather than crash out of the European Union on 29 March".
"Some of our group just feel that they have been forced into no other option but to vote for some kind of delay or pause simply because they don't want to see no deal", said Conservative lawmaker Simon Hart, who heads the Brexit Delivery Group made up of both pro-EU and pro-Brexit Conservatives.
Juncker said failure to agree an orderly divorce would be economically devastating.