The DUP's Jeffrey Donaldson has said there is "no chance" his party would accept an arrangement which allowed Northern Ireland voters to elect MEPs in the Republic of Ireland after Brexit.
The suggestion came as the European Parliament's Brexit negotiator Guy Verhofstadt was speaking to a parliamentary committee that is considering how to redistribute the seats given up by the UK.
At present, Northern Ireland elects three MEPs, while the Republic of Ireland elects 11.
Mr Verhofstadt said those with Irish passports in Northern Ireland should still have a vote in European elections across the border.
He said it could be done by increasing the number of European parliamentary seats in the Republic of Ireland.
Committee chair Danuta Hubner, said: "It is amazing what you are saying, because it is the Republic of Ireland which is against this idea of giving the right to vote to nationals in third countries - so we have a problem here."
In addition, DUP MP Sir Jeffrey Donaldson said the 10 DUP MPs, who are propping up Theresa May's minority government, would insist to her that the proposal be rejected. He said: "We would be strongly opposed to his suggestion. Not least on the grounds that it breaches the Good Friday Agreement and the constitutional settlement that was reached. While, of course, people in Northern Ireland have the right to carry an Irish passport - but to then suggest that Irish passport holders should have the right to vote in another jurisdiction would be a breach of constitutional arrangements agreed. This idea would upset the delicate constitutional balance we have worked out here and would endanger the peace process."