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The news Tuesday came a day after Britain's minister for Brexit talks Stephen Barclay and Attorney General Geoffrey Cox met EU negotiator Michel Barnier.
The Prime Minister is trying to secure changes to the Irish backstop, an insurance policy designed to avoid a hard border between Northern Ireland and the Irish Republic after Brexit, the BBC reported
May lost the vote by 303-258, weeks after she asserted that she had a "substantial majority" of MPs in favour of her Brexit plan.
Hardline pro-Brexit lawmakers say a measure to be voted on Thursday rules out the threat of leaving the EU without an agreement on future relations, undermining Britain's bargaining position.
May is set to update parliament on her latest meetings in Brussels and Dublin aimed at securing a divorce agreement with the EU, with Britain due to leave the bloc on March 29.
They reckoned that some 103,000 jobs would be under threat in Europe's largest economy Germany and 50,000 in France.
While British PM Theresa May said that it is not going to be easy, the Brexit will happen on time.
Ireland's Varadkar met Tusk, who represents EU member states as president of the European Council, before talks with EU Commission president Jean-Claude Juncker.
Following an "inconclusive" meeting with British MPs, British officials feel that PM May might be able to get a deal if EU makes changes to the agreement.
MPs voted last week to send May back to Brussels to renegotiate the clause, suggesting her deal would then be able to pass after it was roundly rejected in parliament last month.
Foreign Secretary Jeremy Hunt felt that delay will occur if negotiations on the deal are dragged on.
At Prime Minister's Questions, Corbyn repeatedly urged May to rule out a no-deal Brexit and to ditch her "red lines" in any upcoming talks with Brussels.
MPs late Tuesday voted through an amendment saying they would only support a divorce deal if its controversial "backstop" clause to keep the Irish border open was removed.
In a series of votes, lawmakers rejected a no-deal Brexit by 318 votes to 310, undermining the government's argument that Britain would be willing to crash out of the EU without an agreement.
Legislators will vote on proposals that have been submitted by both pro-Brexit and pro-EU legislators on Tuesday.