The agreement will also see Northern Ireland remain within the single market in a bid to avoid a hard border on the island of Ireland, should both sides fail to secure a trade deal after the transition period.
Earlier on Thursday, May told Northern Ireland journalists that Irish border talks would likely continue until November.
Ahead of the critical European summit next week, some cabinet members want any backstop arrangements - which could place all of the United Kingdom in a customs union after the end of the Brexit transition period in 2020 - to be clearly time-limited.
A day earlier, the DUP had said it was prepared to vote down the budget if the prime minister in their eyes conceded too much ground to Brussels in the European Union talks, risking an abrupt end to the confidence and supply agreement that has kept May in Downing Street since the 2017 election in which the Tories lost their overall majority.
Barnier said Brexit would trigger the need for customs, Value-Added Tax and compliance checks with EU standards between Ireland and Northern Ireland in the event that a planned "backstop" were triggered because a future EU-UK trade deal was not sufficient in itself to ensure the land frontier was not a "hard border".
But he made clear that the European Union envisages a deal involving new customs and regulatory checks on goods travelling from Britain to Northern Ireland, including health and sanitary inspections for 100% of animals and animal products arriving in ports like Larne and Belfast from the rest of the UK.More news: Will Smith Debuts New Tease for Disney's Aladdin
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The Northern Irish party that British Prime Minister Theresa May's government relies on for support says it will consider backing a vote of no-confidence if May agrees to European Union checks on goods entering the region post-Brexit. "That was the case on Monday and it remains the case today", the prime minister's official spokesman said. Many of her cabinet colleagues have assured me of their Unionism.
He reiterated the EU's line welcoming May's proposals for a bold free-trade deal "without tariffs or quotas" and close security ties after Brexit.
The DUP has threatened to vote against the government's budget if it goes ahead with the plan.
"But that's not a question for us, we're not members of the Conservative Party".
The Prime Minister relies on the support of 10 of Northern Ireland's DUP MPs to prop her up in the House of Commons and to secure their backing senior Tories are ready to offer them a bribe.
"Without an end date, we could be in the customs union forever".