She told a meeting of businessmen and women in Belfast that there were "a number of ways" to amend the amend the backstop, which the Biritsh parliament has demanded before agreeing a divorce deal with Europe.
She also emphasized the government's support for the Good Friday agreement, the 1998 treaty that largely ended decades of violence in Northern Ireland known as "the Troubles".
British Prime Minister Theresa May is set to travel to Brussels on Thursday to discuss the stalled Brexit deal with EU Commission President Jean-Claude Juncker.
It follows a day in which the Prime Minister faced hard questions on the economy, as new data indicated the uncertainty over Brexit could be leading to a slowdown.
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"I think to extend simply because we hadn't reached an agreement would not provide any impetus for that agreement to be reached", said Fox, adding, "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".
A divorce deal between Britain and the European Union is in danger because U.K lawmakers have rejected a section of the agreement meant to ensure there is no hard border between the U.K.'s Northern Ireland and EU member Ireland.
The committee's Conservative vice chair, John Whittingdale, told reporters that he had given Selmayr a suggestion that the backstop be removed from the deal.
A source from the UK's Cabinet Office said: "These emergency evacuation plans have been in existence since the Cold War, but have now been repurposed in the event of civil disorder following a no-deal Brexit".
The Northern Irish Democratic Unionist Party, which props up May's government, said it wanted to get a deal agreed but made clear the border backstop had to be replaced.
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But her meeting with Arlene Foster, the DUP leader, will be the most closely scrutinised given the Prime Minister.
The Prime Minister risked infuriating hardline Brexiteers on Tuesday after ruling out removing the controversial Irish border protocol from her deal as she said she would focus on making changes to it instead. "I don't want to see him again", Puissesseau told the Daily Telegraph.
The border between the Irish Republic and Northern Ireland will be the UK's only land border with the European Union after Brexit, now scheduled on March 29.
Local businesses have urged the government to accept the backstop and avoid a disastrous no-deal Brexit leading to a hard border - which is what the United Kingdom is heading towards, while it attempts to renegotiate a backup plan with the European Union that aims at avoiding precisely that scenario.
In Brussels, a group of British lawmakers met the head of the European Union civil service and said Martin Selmayr appeared to indicate that the European Union might bind itself to new legal conditions.
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As the risk of Britain leaving the European Union without an amicable withdrawal agreement continues to hang over ongoing negotiations, emergency proposals to rescue the royal family during the Cold War era have been "repurposed" in recent weeks to ensure their protection.