"The prime minister remains very clear that she thinks that a very major economy like the United Kingdom needs to have the freedom to be able to make its own trade deals, so she's disagreeing with Jeremy Corbyn's suggestion that we enter a permanent customs union", he told the broadcaster.
Liam Fox, the International Trade Secretary, described Labour's plan as a "dangerous delusion", while Boris Johnson, the former foreign secretary, accused Mr Corbyn of trying to trap Mrs May into a "toxic" Brexit.
The EU says the backstop is vital to avoiding the return of border controls in Ireland and has refused to reopen the Brexit divorce deal, though May insists she can get legally binding changes to replace the most contentious parts of the backstop.
Theresa May sent Mr Barclay and her top European advisor Olly Robbins to hold official talks for the first time since the draft Brexit withdrawal agreement was signed off by European leaders in November a year ago.
Asked if changes to the backstop proposals could come in a separate codicil to the Withdrawal Agreement, Mr Johnson said: "I don't think that would be good enough". Britain is due to leave the European Union on March 29, but lawmakers in Parliament have emphatically rejected Mrs May's divorce deal, raising the prospect of a no-deal exit from the bloc unless more parliamentary support is found.
Corbyn had earlier laid out five Labour Party conditions for supporting any revised Brexit deal. 'Non-EU members do not have a say in EU trade policy so to pretend that you could do so is a unsafe delusion, ' he said.
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"That gives that sense of timetable, clarity, and objective on what we're doing with the European Union - taking that work forward and our determination to get a deal - but equally knowing that role that parliament very firmly has", Brokenshire told the BBC. "Since markets are linked up across the globe and suppliers based outside the European Union are also affected by a no deal Brexit", they said.
The existing Political Declaration, setting out the goals for the future relationship between the United Kingdom and the EU, "explicitly provides for the benefits of a customs union - no tariffs, fees, charges or quantitative restrictions across all sectors and no checks on rules of origin", Mrs May told Mr Corbyn.
She also questioned whether the call for "frictionless" trade would mean reneging on Labour's commitment to end free movement.
Mr Barclay will later travel to Brussels for talks over dinner with the EU's chief negotiator Michel Barnier.
"The government will commit that if the meaningful vote, in other words the deal coming back, has not happened by the 27th of February, then we would allow a further motion - votable in Parliament - to take place, to give that sense of assurance as to the process moving forward", he told the BBC's "Andrew Marr Show".
Her response comes ahead of another week of Brexit debate and voting in Westminster.