The British government's insistence on the stance is increasingly at odds with Irish demands for guarantees, ahead of a mid-December summit when EU leaders will decide if Brexit talks can move on to trade.
The impasse threatens to scupper hopes in London that "sufficient progress" has been made on the border, as well as citizens' rights and its divorce bill to leave the bloc, to move the negotiations into the next phase.
"But we can't get a final answer to the Irish question until we get an idea of the end state -- and until we get into discussions with the EU on the end state that will be very difficult."
He added: "So the quicker we can do that the better."
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Fox's comments came as Phil Hogan, the Irish Republic's EU agriculture commissioner, restated his country's threat to block Brexit talks progress unless it receives firm assurances of no hard border on the island.
Hogan told a British Sunday newspaper that May's ministers had "blind faith" in reaching a comprehensive free trade deal, and warned Dublin would "continue to play tough to the end" over the border.
Arlene Foster, the leader of Northern Ireland's Democratic Unionist Party -- which props up May's minority Conservative government in Westminster -- also weighed in, saying her party opposed any internal trade barrier within the UK. She also accused the Irish government of intransigence.