Irish Prime Minister Leo Varadkar has called for "unique solutions" to relations between Britain and the European Union after Brexit, including a possible bilateral customs union between Britain and the EU and an alternative to the European Court of Justice to oversee any deal.

Ireland, which after Brexit will have the EU's only land border with the United Kingdom, is widely seen as the EU country most exposed to the fall-out from Brexit.

Speaking during his first visit to Northern Ireland since his appointment as prime minister in June, Varadkar said Britain and the EU would need "unique solutions" if we are to avoid a damaging breakdown in trade.

"Perhaps there can be an EU-UK customs union. After all, we have one with Turkey. Surely we can have one with the United Kingdom?" Varadkar said in a speech at Queen's University in Belfast.

Varadkar said it was not yet clear if enough progress had been made by a summit in October to move on to talks about a free trade deal.

He said one question was whether it would be possible to find an alternative to the European Court of Justice to oversee any future deal between Britain and the EU.