MPs voted 317 to 301 that the Irish backstop should be removed from the Brexit withdrawal agreement and replaced with alternative arrangements.
"I will be armed with a fresh mandate, new ideas and a renewed determination to agree a pragmatic solution that delivers the Brexit the British people voted for, while ensuring there is no hard border between Northern Ireland and the Republic of Ireland", she wrote.
British Prime Minister Theresa May said she would seek a "pragmatic solution" to the impasse over Brexit when she tries to reopen talks with Brussels in the coming days.
"The backstop is a necessary guarantee, based on legal certainty, not just wishful thinking", he wrote in the Sunday Times.
He is the second senior minister to suggest such a delay may be needed, after Foreign Secretary Jeremy Hunt said on Thursday Britain may need time to get legislation through.
"It's 100 per cent untrue", a Downing Street spokeswoman said about the article, which reported that May's advisors were considering a June 6 election under a scenario in which Brexit was delayed past March 29 but May subsequently won parliament's backing for a withdrawal deal in April.
May has promised MPs that she will bring any revised deal back to be voted on by MPs on February 13.
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Under the terms of the deal, the United Kingdom would remain in the customs union until an alternative arrangement is found that ensured the avoidance of a hard border.
If such an agreement could not be reached, then to avoid those checks and border posts or other infrastructure, the backstop would come into force.
Tory MPs have called for the removal of the backstop from the Brexit deal or for the agreement to include a time limit or a unilateral exit mechanism.
Our political correspondent Chris Mason said that while Mrs May pledged to "go back to Brussels to secure a plan that Parliament can stand behind", the European Union remains publicly opposed to changing the backstop.
"Now the PM co-opts us into accepting everything but the backstop", he said on Twitter, adding if all that was agreed on the backstop was a codicil, May could "expect a further substantial defeat for the agreement".
Members of parliament, who last month rejected May's Brexit deal with the European Union, have instructed her to return to Brussels to renegotiate the arrangements for the post-Brexit border between the United Kingdom and Ireland.
"There is a big difference between if we had an agreement and we need some time to get the legalities done, that is one thing", Fox said.