"The UK is clear that the future customs arrangement needs to deliver on the commitments made in relation to Northern Ireland".
The strength in opinion carried across political divides.
Mr Juncker will be accompanied by European Union agriculture commissioner Phil Hogan and Mr Barnier.
The EU says Britain's decision to leave the single market and the customs union means border checks will be necessary unless Northern Ireland retains the same customs and regulations as the EU.
Guy Verhofstadt said the government's position that the backstop of continuing United Kingdom alignment with the EU customs union would cease by the end of 2021 if no other solution was found left him "puzzled".
Following the meetings, Juncker made it clear that a resolution to the border issue was crucial to the withdrawal treaty.
Across Great Britain, 62 per cent of respondents said Brexit negotiations were proceeding too slowly, with 30 per cent blaming the delays on "politicians in Britain who want a soft Brexit or to stop Brexit altogether".
LONDON - Irish Taoiseach Leo Varadkar has warned Theresa May that "there isn't much time left" to find a solution to the Irish border dilemma which holds the key to a final Brexit deal.
Britain's plan for a temporary customs "backstop" is "not acceptable" and any fall-back option can not be time-limited, the European Parliament's chief Brexit co-ordinator has told MPs.
Jean-Claude Juncker will discuss Brexit ahead of this month's European Council meeting of leaders in Brussels.
Juncker said he was "strongly against any temptation" to try to isolate Ireland and not to conclude a withdrawal deal.
"The Withdrawal Agreement without a backstop is of no use to us whatsoever".
"This is an important opportunity to assess the state of play in the Brexit negotiations".
"The backstop paper has been amended and now expresses, in much more detail, the time limited nature of our proposal - something the prime minister and David Davis have always been committed to".
But he said that no deal is as good as membership of the EU.
Later, Mr Juncker received an honorary degree from the National University of Ireland.
The Brexiteer Democratic Unionists criticised Mr Juncker's comments on Brexit, accusing the European Union of using Ireland as a "bargaining chip" in the negotiations.
"The Republic of Ireland is being used as a bargaining chip by the EU".
Brexit talks are now at an impasse over the "backstop" - the fallback arrangement if negotiations over a future trade deal between Britain and European Union fail to solve the Irish border question.
Tomorrow he will meet Irish President Michael D Higgins and visit the home of the Gaelic Athletic Association (GAA), Croke Park, and the GAA museum.