Britain's Brexit Secretary Stephen Barclay said "the dynamic did change" with the vote, but offered no specifics on what the alternative arrangements might involve, saying London was looking at technological solutions, time limits and exit clauses.
Conservative MPs have been told that the ten-day break from February 15 will be cancelled but they may still be able to return to their constituencies.
Legislation needs to be put before the House for a minimum number of sitting days between stages and the extra week would give Theresa May breathing room as she prepares for talks with European leaders.
The EU parliament's point-man on Brexit, Guy Verhofstadt, underlined that nobody in Europe wanted to use the backstop, but that it's "needed to be 100 per cent sure that there is no border between Northern Ireland and the Irish republic". "We have tested them and we have found that they do not stand up to scrutiny, and now we have a British prime minister advocating again for the same things that were tested", Coveney said. The vote was not legally binding, but has political force as an expression of the will of parliament.
A no-deal Brexit might have been staved off by another amendment that would have compelled the British government to ask the European Union for an extension of the Brexit date, but lawmakers rejected it.
He said Brussels was open to a delay beyond the March 29 deadline, but would continue its preparations for a no-deal Brexit on March 29.
It courts Brexiteers with a promise to ditch an unpopular Irish border policy in favour of an alternative, and appeals to remain-supporting Conservatives by pledging safeguards against the risk of disruption if no deal can be agreed with the EU.
Juncker told European Union lawmakers that both sides had already discussed the UK's demands to seek alternative arrangements to the Irish backstop, a provision to try to keep an open border between Ireland and the United Kingdom province of Northern Ireland.
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"The backstop is part of the Withdrawal Agreement, and the Withdrawal Agreement is not open for renegotiation", the spokesman said. This could be an oppportunity for an eleventh-hour deal, or it would be the last chance to agree an extension of the Article 50 negotiation period and delay Brexit to avoid no-deal disruptions.
While the European Union has repeatedly refused to reopen the divorce deal, European Union sources said additional clarifications, statements or assurances on the backstop might be possible.
Prior to the vote, Mr Varadkar told the Dáil (Irish Parliament) he expected to speak to Mrs May after the vote - although an Irish government spokesperson said the call would likely be later on Wednesday.
Buoyed by winning a vote in parliament, May has vowed to secure "legally binding changes" to the withdrawal agreement.
Juncker called on British lawmakers to make clear what they want.
And there was no sign that European leaders were prepared to unpick the backstop in order to salvage the deal. For too long it has been too easy to say what we don't like and to expect the other side to accede to everything we want.
"The House of Commons could have asserted itself tonight - instead it indulged the Prime Minister's decision to chase a fairy tale", the First Minister tweeted.
Jill Lawless reported from London.