Writing in the Sunday Telegraph, she suggested that Labour leader Jeremy Corbyn had agreed with her that the Irish border backstop as set out in the Withdrawal Agreement had to change. The EU has said the deal can not be renegotiated.
The prime minister added that MPs wanted the government to go back to Brussels to renegotiate the deal after the Commons voted in favour of Tory backbencher Sir Graham Brady's amendment on Tuesday that called for "alternative arrangements" to be found.
MPs vote that the Irish backstop should be removed from the Brexit withdrawal agreement.
"She Wrote When I return to Brussels I will be battling for Britain and Northern Ireland, I will be armed with a fresh mandate, new ideas and a renewed determination to agree a pragmatic solution".
British Prime Minister Theresa May said on Sunday she would seek a "pragmatic solution" to a parliamentary impasse over the terms on which Britain leaves the European Union when she tries to reopen talks with Brussels.
May said she was "determined" to deliver Brexit on March 29, but Fox echoed comments by Foreign Secretary Jeremy Hunt on Thursday that a delay to that leaving date may be needed in order to get legislation through parliament.
Warnings of the consequences of a no-deal Brexit intensified last week as several multinational firms, including Airbus and Ford, said jobs could suffer with an unfavourable negotiating outcome.
Mr Coveney said while MPs had urged Mrs May to replace the backstop with an alternative arrangement, no one has succeeded in putting forward a "credible" second option.
Opponents to Brexit were quick to seize on the Sky News claim, which the firm said it would not comment on.
May's office denied a report in the Mail on Sunday newspaper that the prime minister's advisors were considering scheduling a national election for June 6.
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.
On a visit to Stoke-on-Trent on Saturday, he said: "I don't think any MP will sell their votes in that way - that sort of bribery and corruption". "There must not be a hard border to the Republic of Ireland".
Under the backstop, the United Kingdom would remain in a customs union with the EU in the absence of a trade deal, and Northern Ireland would stay aligned to some EU rules. "There is a deal on the table, and walking away from it will have serious consequences for us all".